Double-Layer Potentials, Configuration Constants, and Applications to Numerical Ranges

Abstract

Given a compact convex planar domain $\mathrm{\Omega }$ with non-empty interior, the classical Neumann’s configuration constant $c_{\mathbb{R}}(\mathrm{\Omega })$ is the norm of the Neumann–Poincaré operator $K_{\mathrm{\Omega }}$ acting on the space of continuous real-valued functions on the boundary $\partial \mathrm{\Omega }$⁠, modulo constants. We investigate the related operator norm $c_{\mathbb{C}}(\mathrm{\Omega })$ of $K_{\mathrm{\Omega }}$ on the corresponding space of complex-valued functions, and the norm $a(\mathrm{\Omega })$ on the subspace of analytic functions. This change requires introduction of techniques much different from the ones used in the classical setting. We prove the equality $c_{\mathbb{R}}(\mathrm{\Omega })=c_{\mathbb{C}}(\mathrm{\Omega })$⁠, the analytic Neumann-type inequality $a(\mathrm{\Omega })<1$⁠, and provide various estimates for these quantities expressed in terms of the geometry of $\mathrm{\Omega }$⁠. We apply our results to estimates for the holomorphic functional calculus of operators on Hilbert space of the type $\Vert p(T)\Vert \le K\underset{z\in \mathrm{\Omega }}{sup}|p(z)|$⁠, where $p$ is a polynomial and $\mathrm{\Omega }$ is a domain containing the numerical range of the operator $T$⁠. Among other results, we show that the well-known Crouzeix–Palencia bound $K\le 1+\sqrt{2}$ can be improved to $K\le 1+\sqrt{1+a(\mathrm{\Omega })}$⁠. In the case that $\mathrm{\Omega }$ is an ellipse, this leads to an estimate of $K$ in terms of the eccentricity of the ellipse.

1 Introduction

1.1 Double-layer potential

Throughout this article, $\mathrm{\Omega }$ will denote a compact convex planar domain with non-empty interior. If $C(\partial \mathrm{\Omega })$ is the space of continuous functions on the boundary $\partial \mathrm{\Omega }$ and $f\in C(\partial \mathrm{\Omega })$⁠, then its double-layer potential  $u$ is the harmonic function

Here $ds=|d\sigma |$ is the arclength measure on the rectifiable curve $\partial \mathrm{\Omega }$⁠, ${\mathrm{\Omega }}^o$ is the interior of $\mathrm{\Omega }$⁠, and $N(\sigma )$ is the outer-pointing normal at the boundary point $\sigma$⁠. The equality between the two expressions for $u(z)$ above follows from an elementary computation in the case that $\partial \mathrm{\Omega }$ is sufficiently smooth. In the general case, we interpret $N(\sigma )(\sigma -z{)}^{-1}$ as a Borel measurable function on $\partial \mathrm{\Omega }$⁠. By convexity of the domain, both the tangent $T(\sigma )$ and the normal $N(\sigma )$ exist and are continuous at all but a countable number of points $\sigma$⁠, which we will call corners, at which the discontinuity of $T$ and $N$ amounts to a jump in the argument. In Appendix  A we include more details regarding boundaries of planar convex domains, and other facts mentioned below.

The Neumann–Poincaré operator appears in connection with the study of boundary behaviour of the double-layer potential. It is known that $u$ given by (1) has a continuous extension to $\partial \mathrm{\Omega }$⁠, and we have the representation

where $K_{\mathrm{\Omega }}$ denotes the Neumann–Poincaré integral operator

Here ${\mu }_{\zeta }$ is the probability measure

where ${\theta }_{\zeta }$ can be interpreted as the angle of the aperture at the possible corner at $\zeta$ of $\partial \mathrm{\Omega }$⁠, ${\delta }_{\zeta }$ is a unit mass at the point $\zeta$⁠, and ${\rho }_{\zeta }$ is the Radon–Nikodym derivative

It is natural to use the convention that ${\theta }_{\zeta }=\pi$ if $\zeta$ is not a corner. This occurs precisely when ${\mu }_{\zeta }$ assigns no mass to the singleton $\{\zeta \}$⁠. We will say that the collection of measures $\{{\mu }_{\zeta }{\}}_{\zeta \in \partial \mathrm{\Omega }}$ is the Neumann–Poincaré kernel of $\mathrm{\Omega }$⁠.

The density ${\rho }_{\zeta }$ has the following useful geometric interpretation. If $\sigma \in \partial \mathrm{\Omega }\setminus \{\zeta \}$ is not a corner, and $R_{\zeta ,\sigma }$ is the radius of the unique circle passing through $\zeta$ that is tangent to $\partial \mathrm{\Omega }$ at $\sigma$⁠, then the equality

holds. The radius $R_{\zeta ,\sigma }$ may degenerate to $\mathrm{\infty }$ if $\zeta$ is contained in the tangent line to $\partial \mathrm{\Omega }$ passing through $\sigma$⁠. In that case we see easily that ${\rho }_{\zeta }(\sigma )=0$⁠, so (5) still holds. To establish the formula, note that the center $m$ of the circle in question is of the form $m=\sigma -RN(\sigma )$⁠, where the radius $R=R_{\zeta ,\sigma }>0$ of the circle satisfies $|m-\zeta {|}^2=|(\sigma -\zeta )-RN(\sigma ){|}^2=R^2$⁠. Expanding the squares and solving for $R$ leads to (5).

1.2 Neumann’s configuration constant

1.2.1 Real configuration constant

Historically, the Neumann–Poincaré operator has been used to solve the Dirichlet problem of finding a harmonic extension to ${\mathrm{\Omega }}^o$ of a given continuous function $u$ on $\partial \mathrm{\Omega }$⁠. The extension can be obtained by finding $f\in C(\partial \mathrm{\Omega })$⁠, which solves (2). Indeed, if such an $f$ is found, then the extension of $u$ to ${\mathrm{\Omega }}^o$ is given by the double-layer potential in (1). This naturally leads to questions of invertibility of the operator $I+K_{\mathrm{\Omega }}$ appearing on the right-hand side of (2), and consequently to the introduction of the Neumann’s configuration constant, which we shall soon define as the operator norm of $K_{\mathrm{\Omega }}$ acting on an appropriate space. Note that if $\mathbf{1}$ is the constant function, then we have that $K_{\mathrm{\Omega }}\mathbf{1}=\mathbf{1}$⁠, since each ${\mu }_{\zeta }$ is a probability measure. Thus $K_{\mathrm{\Omega }}$ can be naturally defined as a linear mapping on the quotient space $C(\partial \mathrm{\Omega })/\mathbb{C}\mathbf{1}$⁠. The classical approach is to instead consider $K_{\mathrm{\Omega }}$ as acting on the space of real-valued continuous functions $C_{\mathbb{R}}(\partial \mathrm{\Omega })$⁠, in which case the corresponding quotient space $C_{\mathbb{R}}(\partial \mathrm{\Omega })/\mathbb{R}\mathbf{1}$ is endowed with the norm

It is not hard to see that the two above expressions for the norm of the coset $g+\mathbb{R}\mathbf{1}$ are equivalent: they are both equal to half of the length of the interval $g(\partial \mathrm{\Omega }):=\{g(\partial \mathrm{\Omega }):\zeta \in \partial \mathrm{\Omega }\}$⁠, the image of $g$⁠. The right-most expression is minimized by choosing $r$ to be the mid-point of the image interval. Neumann’s (real) configuration constant $c_{\mathbb{R}}(\mathrm{\Omega })$ is defined as the operator norm of $K_{\mathrm{\Omega }}$ acting on the quotient space $C_{\mathbb{R}}(\partial \mathrm{\Omega })/\mathbb{R}\mathbf{1}$⁠:

It is not hard to see that we may let $K_{\mathrm{\Omega }}$ instead act from $C_{\mathbb{R}}(\partial \mathrm{\Omega })$ into the quotient $C_{\mathbb{R}}(\partial \mathrm{\Omega })/\mathbb{R}\mathbf{1}$ without affecting the operator norm. Since each measure ${\mu }_{\zeta }$ is of unit mass, we have $0\le c_{\mathbb{R}}(\mathrm{\Omega })\le 1$⁠. If

then

where we use the total variation norm (functional norm) on the right-hand side. By varying $f$ over the unit ball of $C_{\mathbb{R}}(\partial \mathrm{\Omega })$ and $\zeta ,{\zeta }^{\prime }$ over $\partial \mathrm{\Omega }$⁠, we obtain the important relation

This expression for $c_{\mathbb{R}}(\mathrm{\Omega })$ will play a fundamental role in our study.

1.2.2 Neumann’s lemma

From (8) we can immediately deduce that $c_{\mathbb{R}}(\mathrm{\Omega })=1$ in the case that $\mathrm{\Omega }$ is a triangle or a convex quadrilateral. Indeed, in those cases one sees from (3) and (4) that if ${\zeta }_1$ and ${\zeta }_2$ are corners of $\mathrm{\Omega }$ (opposing, in the case of the quadrilateral) then ${\mu }_{{\zeta }_1}$ and ${\mu }_{{\zeta }_2}$ are mutually singular, and so $\Vert {\mu }_{{\zeta }_1}-{\mu }_{{\zeta }_2}\Vert =2$⁠, implying $c_{\mathbb{R}}(\mathrm{\Omega })=1$⁠. Neumann’s lemma, which appears initially in Neumann’s book [14], states that the cases of the triangle and quadrilateral are exceptional. For any other type of domain we have the strict inequality $c_{\mathbb{R}}(\mathrm{\Omega })<1$⁠. See [17] for a proof of this claim by Schober, and the curious history of incomplete attempts at a valid proof in full generality. Neumann’s lemma implies the invertibility of $I+K_{\mathrm{\Omega }}$ on $C_{\mathbb{R}}(\partial \mathrm{\Omega })/\mathbb{R}\mathbf{1}$⁠, and thus the solvability of the Dirichlet problem on a convex domain $\mathrm{\Omega }$⁠, which is not one of the two exceptional cases. The remaining cases can be handled by considering instead powers of $K_{\mathrm{\Omega }}$⁠. See, for instance, [13, Theorem 3.8], [6, Proposition 7], or the article [16], which contains also an exposition of the double-layer potential and Neumann’s lemma.

At the other extreme, we have $c_{\mathbb{R}}(\mathrm{\Omega })=0$ if and only if $\mathrm{\Omega }$ is a disk. This result will be proved in Section 5.

1.3 Complex and analytic configuration constants

1.3.1 Two new configuration constants

In the present article we will discuss certain applications of the double-layer potential to operator theory, which motivate the definition of the complex configuration constant

The difference between (7) and (9) is that the latter is the norm of $K_{\mathrm{\Omega }}$ on the larger space of complex-valued functions. As a consequence, we have $c_{\mathbb{R}}(\mathrm{\Omega })\le c_{\mathbb{C}}(\mathrm{\Omega })\le 1$⁠. There is a principal difference between the geometric interpretations of the norms in the quotient spaces $C_{\mathbb{R}}(\partial \mathrm{\Omega })/\mathbb{R}\mathbf{1}$ and $C(\partial \mathrm{\Omega })/\mathbb{C}\mathbf{1}$⁠. In the former case, as we have already noted, the norm (6) of the coset represented by the real-valued function $g$ is equal to half of the length of the image of $g$⁠, this image being an interval on the real line $\mathbb{R}$⁠. In the case of complex-valued $g$⁠, the quotient norm

can instead be interpreted as the radius of the smallest disk containing the image of $g$. A crucial difference is that we lose the ability to estimate the norm of the coset $g+\mathbb{C}\mathbf{1}$ by considering the quantities $|g(\zeta )-g({\zeta }^{\prime })|$ only. This is the essence of why new tools are required to treat this case.

We will also study an analogous analytic constant, which is the norm of the operator $K_{\mathrm{\Omega }}$ restricted to the subspace of analytic functions in $C(\partial \mathrm{\Omega })$⁠. More precisely, we let $\mathcal{A}(\mathrm{\Omega })$ be the space of functions that are continuous in $\mathrm{\Omega }$ and analytic in ${\mathrm{\Omega }}^o$⁠. Each function in $\mathcal{A}(\mathrm{\Omega })$ has a unique restriction to $\partial \mathrm{\Omega }$⁠, and thus $\mathcal{A}(\mathrm{\Omega })$ can be naturally identified with a subspace of $C(\partial \mathrm{\Omega })$⁠. We define the analytic configuration constant as

The space $\mathcal{A}(\mathrm{\Omega })$ is not invariant under $K_{\mathrm{\Omega }}$⁠, but we do have that $K_{\mathrm{\Omega }}f$ is the complex conjugate of a function in $\mathcal{A}(\mathrm{\Omega })$ (in [5, proof of Lemma 2.1] this claim is established for $\mathrm{\Omega }$ with smooth boundary, but the same argument works in general). Clearly, we have the inequality $a(\mathrm{\Omega })\le c_{\mathbb{C}}(\mathrm{\Omega })$⁠. We note also that if $\tilde{\mathrm{\Omega }}$ is the image of $\mathrm{\Omega }$ under an affine transformation of the plane, then the configuration constants of the two domains are equal. We shall verify this claim in Section 6.

1.3.2 An application to functional calculi

Given an operator $T$ on a Hilbert space $\mathcal{H}$ with numerical range

we are interested in the optimal constant $K>0$ in the inequality

where $p$ is an analytic polynomial, and the left-hand side is the operator norm of $p(T)$ acting on $\mathcal{H}$⁠. More generally, if $W(T)$ in (12) is replaced by an arbitrary domain $\mathrm{\Omega }$⁠, and if the corresponding inequality holds for some $K$⁠, then we say that $\mathrm{\Omega }$ is a $K$-spectral set for $T$⁠. Von Neumann’s inequality says that the unit disk is a $1$-spectral set for any contraction $T$⁠, and a result of Okubo–Ando from [15] says that any disk containing $W(T)$ is a $2$-spectral set for $T$⁠.

The numerical range $W(T)$ is a bounded convex subset of the plane, its closure $\overline{W(T)}$ contains the spectrum $\sigma (T)$ of $T$⁠, and it has non-empty interior in the case that $T$ is not a normal operator (see, for instance, [10, Chapter 1]). For normal operators, the bound (12) with constant $K=1$ is a consequence of the spectral theorem, and it suffices to take the supremum on the right-hand side over the smaller set $\sigma (T)$⁠. For general $T$⁠, even establishing the existence of a bound as in (12) is a non-trivial task. A result of Delyon–Delyon from [6, Theorem 3] establishes the existence of the bound, and shows that $K$ can be chosen depending only on the area and the diameter of $W(T)$⁠. The remarkable work of Crouzeix in [3] establishes that (12) holds with $K\le 11.08$⁠. A subsequent work of Crouzeix and Palencia in [5] improves the estimate to $K\le 1+\sqrt{2}$⁠. The Neumann–Poincaré operator appears as an essential tool in all of the mentioned works. The standing conjecture of Crouzeix from [2] is that the bound holds with $K=2$⁠. This bound is presently known to hold in the case $\mathcal{H}$ being of dimension 2, and has been established by Crouzeix in [2].

Our interest in the new notions of configuration constants is inspired by a recent work of Schwenninger and de Vries in [18], where bounds for general homomorphisms between uniform algebras and the algebras of bounded linear operators are studied. In Section 6 we will combine their arguments with the methods of Crouzeix–Palencia to obtain the following estimate:

For instance, if $W$ is a disk, then $a(W)=0$⁠, which gives the Okubo–Ando result mentioned above. In [18], Schwenninger and de Vries recovered this result also. The estimate (13) is our motivation for the following investigation of the configuration constants $c_{\mathbb{R}}(\mathrm{\Omega }),c_{\mathbb{C}}(\mathrm{\Omega })$⁠, and $a(\mathrm{\Omega })$⁠, and the relations between them.

1.4 Main results

1.4.1 Relation between the real and complex constants

Consider the situation in Figure 2, where the triangular image of the complex-valued function $g:\partial \mathrm{\Omega }\to \mathbb{C}$ is contained in a disk of radius $1$⁠, and intersects the boundary circle of the disk in three distinct points. The three-point set $\{g({\zeta }_1),g({\zeta }_2),g({\zeta }_3)\}$ is not contained in any open half-circle of the boundary, and it follows from a simple geometric argument (which we shall present in the proofs below) that $\Vert g+\mathbb{C}\mathbf{1}{\Vert }_{\partial \mathrm{\Omega }}=1$⁠. However, the sides of the triangular image of $g$ are all of lengths strictly less than $2$⁠, and this implies that

If such a function $g$ lies in the image of the unit ball of $C(\partial \mathrm{\Omega })$ under the Neumann–Poincaré operator $K_{\mathrm{\Omega }}$ for some domain $\mathrm{\Omega }$ that satisfies $c_{\mathbb{R}}(\mathrm{\Omega })<1$⁠, then a strict inequality $c_{\mathbb{R}}(\mathrm{\Omega })<c_{\mathbb{C}}(\mathrm{\Omega })$ occurs. Our first main result excludes this possibility, and so establishes the simplest possible relation between the real and complex configuration constants.

 

It follows that every considered domain has a well-defined configuration constant  $c(\mathrm{\Omega })$⁠, which is equal to the operator norm of $K_{\mathrm{\Omega }}$ on $C(\partial \mathrm{\Omega })/\mathbb{C}\mathbf{1}$⁠, and which can be computed according to the right-hand side of (8). An important consequence of this result is the inequality

which, as we shall soon see, has some interesting consequences.

Theorem 1 doesn’t appear nearly as straightforward to prove as it is to state, and the proof takes up a large portion of the article. However, the only property of the Neumann–Poincaré operator used in the proof is that its integral kernel $\{{\mu }_{\zeta }{\}}_{\zeta \in \partial \mathrm{\Omega }}$ consists of real-valued measures. In fact, the theorem will be deduced as a corollary of a result, which we call the Three-measures theorem, and which is a general statement regarding the geometry of the space $C(X)$ of continuous functions on a compact Hausdorff space $X$⁠. This result, which we discuss and prove in Section 2, puts a restriction on the possible configurations of point sets in the plane, which arise as values of a collection of real-valued functionals on $C(X)$⁠.

1.4.2 Analytic Neumann’s lemma

Note that the above estimate in (14), together with Neumann’s lemma, implies that $a(\mathrm{\Omega })<1$ whenever $\mathrm{\Omega }$ is not a triangle or a quadrilateral. This can be improved, for we have an analytic version of Neumann’s lemma, in which no exceptional cases occur.

 

Our proof of Theorem 2 is much different from the one given by Schober in his proof of the real Neumann’s lemma in [17], but it works also in the real context. At the end of Section 4 we show how our technique leads to a different proof of Neumann’s lemma.

1.4.3 Functional calculus bounds

The following result has already been mentioned above.

 

Recall that if the numerical range of an operator has empty interior, then the operator is normal, and so (12) holds with $K=1$⁠. From this observation and Theorem 2 we obtain that for any fixed operator $T:\mathcal{H}\to \mathcal{H}$⁠, the optimal constant $K$ in (12) is always strictly smaller than $1+\sqrt{2}$⁠. In fact, we deduce from our results that we have the inequality

with a constant

which depends only on the shape of $W=\overline{W(T)}$⁠, and not on the operator $T$ itself. We show in Section 5 that no better universal bound can be obtained by means of the analytic configuration constant: for any $ϵ>0$ there exists a “thin” quadrilateral ${\mathrm{\Omega }}_{ϵ}$ for which we have $a({\mathrm{\Omega }}_{ϵ})>1-ϵ$⁠. However, fixing the dimension of the Hilbert space $\mathcal{H}$⁠, one may combine earlier results of Crouzeix to obtain a uniform improvement. The optimal constant $K$ in (12) varies with $T$⁠, and we may consider the supremum of these quantities among all operators $T$ on a Hilbert space $\mathcal{H}$ of a fixed dimension $N$⁠. In [4, Theorem 2.2], Crouzeix proved that there exists an operator realizing this supremum. An immediate corollary of his result, Theorem 2 and Theorem 3 is the following.

 

This improves the Crouzeix–Palencia bound, although by an indefinite amount.

1.4.4 Estimates for the configuration constants

In Section 5 we present also other computations and estimates for the configuration constants. Surprisingly, in the case of an elliptical domain, the configuration constant is computable exactly, and we obtain

where $a$ and $b$ are lengths of the semi-axes of the ellipse ${\mathrm{\Omega }}_{a,b}$⁠, and $e$ is the eccentricity of the ellipse, given by $e:=\sqrt{1-b^2/a^2}$ in case that $a\ge b$⁠. This fact, together with Theorem 1, estimate (13), and the inequality $a(\mathrm{\Omega })\le c(\mathrm{\Omega })$⁠, has the following consequence.

 

Note that the function $a\mapsto K(a,1)$ is continuous and increasing for $a\ge 1$⁠, and we have

Hence the estimate in Corollary 5 gets worse as the eccentricity of the ellipse ${\mathrm{\Omega }}_{a,b}$ grows, and approaches the Crouzeix–Palencia bound in the limit $a\to \mathrm{\infty }$⁠. On the other hand, as $a\to 1$⁠, the eccentricity of the ellipse ${\mathrm{\Omega }}_{a,1}$ tends to $0$⁠. The estimate is then close to the conjectured optimal bound $K=2$ and coincides with the Okubo–Ando bound for $a=1$⁠, in which case the domain is a disk. From this perspective, Corollary 5 may be interpreted as an elliptical generalization of the Okubo–Ando estimate.

For many other types of domains, the exact value of $c(\mathrm{\Omega })$ is inaccessible. To help the situation, we establish an integral estimate, which gives an upper bound on $c(\mathrm{\Omega })$ in terms of the curvature of $\partial \mathrm{\Omega }$⁠, roughly speaking. For a fixed $\sigma$ that is not a corner of $\partial \mathrm{\Omega }$⁠, recall the definition of $R_{\zeta ,\sigma }$ in (5), and consider

If $\kappa (\sigma )$ is the curvature of $\partial \mathrm{\Omega }$ at $\sigma$⁠, then $R_{\mathrm{\Omega }}(\sigma )$ is at least as large as the radius of curvature

which is also the radius of the osculating circle at $\sigma$⁠. Geometrically, $R_{\mathrm{\Omega }}(\sigma )$ is the radius of the smallest disk tangent to $\partial \mathrm{\Omega }$ at $\sigma$⁠, which contains $\mathrm{\Omega }$⁠, if such a disk exists, and it is equal to $\mathrm{\infty }$ otherwise. The latter case occurs, for instance, if $\sigma$ lies on a straight line segment contained in $\partial \mathrm{\Omega }$⁠. However, if $\partial \mathrm{\Omega }$ is sufficiently curved on a segment of $\partial \mathrm{\Omega }$⁠, then $R_{\mathrm{\Omega }}$ will be bounded above there. We obtain in such a situation a non-trivial upper bound on $c(\mathrm{\Omega })$⁠.

 

The result implies spectral constant estimates similar to the one in Corollary 5 above. It also generalizes some similar results in the literature. See Section 5 for further details and examples.

1.4.5 An unresolved matter

We have mentioned above that $c(\mathrm{\Omega })=0$ if and only if $\mathrm{\Omega }$ is a disk. With some additional effort, we will show in Section 5 that the condition $a(\mathrm{\Omega })=0$ also characterizes disks. In this case, we have the equality $a(\mathrm{\Omega })=c(\mathrm{\Omega })$⁠. It is natural to ask whether other domains exist for which the equality occurs, or if the case of the disk is exceptional.

 

As a consequence of Theorem 2 and the exceptional cases of Neumann’s lemma, we see that the strict inequality holds whenever $\mathrm{\Omega }$ is a triangle or a quadrilateral. The authors have not been able to confirm that the inequality holds in any other examples.

1.5 Notations

Some of our notation has already been introduced above. For a continuous function $f$ defined on a set $X$⁠, we denote by $\Vert f{\Vert }_X$ the supremum of $|f|$ over $X$⁠. For cosets of the form $f+\mathbb{C}\mathbf{1}$ we use the convention

with similar convention for real-valued $f$ and cosets $f+\mathbb{R}\mathbf{1}$⁠. A norm $\Vert \cdot \Vert$ without a subscript usually denotes a linear functional norm or a total variation norm of a measure. The distinction will be unimportant and should anyway be easy to deduce from context. We use boldface letters, such as $\mathbf{x}$⁠, to denote vectors in ${\mathbb{R}}^n$⁠, and plain letters, such as $x_j$⁠, to denote the coordinates.

2 The Three-Measures Theorem

2.1 Definitions of relevant spaces and operators

Theorem 1 will be proved as a corollary of our analysis of three-point configurations

where $x$ is an element of a given normed space $\mathcal{N}$⁠, and ${\ell }_1,{\ell }_2,{\ell }_3\in {\mathcal{N}}^{\ast }$ are three bounded linear functionals on $\mathcal{N}$⁠. A point configuration of this type has to satisfy certain conditions. For instance, we must have the distance bound

Our principal interest will be in estimating the radius of the smallest disk that contains such a three-point set.

In order to use the tools of functional analysis, we will formulate our problem as one of estimating the norm of an operator between normed spaces. To this end, we use the space ${\mathbb{C}}^3$ of triples of complex numbers, and we equip it with the following norm:

Similarly to our previous notational conventions, we shall set $\mathbf{1}:=(1,1,1)\in {\mathbb{C}}^3$⁠. The quotient norm in the quotient space ${\mathbb{C}}^3/\mathbb{C}\mathbf{1}$ satisfies

and it has the geometric interpretation adequate to our problem: it is the radius of the smallest disk containing the three point set $\{a,b,c\}$⁠. Given a normed space $\mathcal{N}$ and three linear functionals ${\ell }_1,{\ell }_2,{\ell }_3\in {\mathcal{N}}^{\ast }$⁠, we introduce the linear operator $\mathcal{L}:\mathcal{N}\to {\mathbb{C}}^3/\mathbb{C}\mathbf{1}$ defined by

With these conventions, each three-point configuration $({\ell }_1(x),{\ell }_2(x),{\ell }_3(x))$ is contained in a disk of radius at most $\Vert \mathcal{L}{\Vert }_{\mathcal{N}\to {\mathbb{C}}^3/\mathbb{C}\mathbf{1}}\cdot \Vert x{\Vert }_{\mathcal{N}}$⁠. We want to estimate the operator norm $\Vert \mathcal{L}{\Vert }_{\mathcal{N}\to {\mathbb{C}}^3/\mathbb{C}\mathbf{1}}$⁠.

2.2 Statement of the theorem

Without any information regarding the space $\mathcal{N}$ or the functionals ${\ell }_1,{\ell }_2,{\ell }_3$⁠, the optimal estimate is

Indeed, we see that we cannot do better by choosing $\mathcal{N}=\mathbb{C}$⁠, $x=1$⁠, and the functionals (scalars) to be the vertices of an equilateral triangle inscribed in the unit circle. For instance,

The sides of the triangle have the common length equal to $|{\ell }_i-{\ell }_j|=\sqrt{3}$⁠, and the smallest disk containing the three points ${\ell }_i(x)={\ell }_i$ is the unit disk itself. Thus, in this case, (19) holds with equality. The estimate holds in general as a consequence of Jung’s theorem, which appeared first in [11], and which in the context of the plane says that any set of diameter $d$ is contained in a disk of radius $d/\sqrt{3}$⁠. In our setting $d\le \underset{j,k}{max}\Vert {\ell }_j-{\ell }_k{\Vert }_{{\mathcal{N}}^{\ast }}$⁠, and so the estimate (19) follows from Jung’s theorem.

In our intended application, the role of the space $\mathcal{N}$ is played by $C(X)$⁠, the Banach space of continuous functions on a compact Hausdorff space $X$⁠, and the functionals are given by integration against real-valued measures

It turns out that the three-point configurations that arise in this way are contained in disks of radius smaller than predicted by Jung’s theorem. The main result of the section is the following.

 

It is the “$\le$” estimate in (20) that is the critical one. The lower bound “$\ge$” follows from the definition of the functional norm

We will spend the rest of the section on proving Theorem 7. The outline of the proof is as follows. We will first use duality to formulate the problem in terms of the adjoint operator ${\mathcal{L}}^{\ast }$ between the dual spaces. Next, a discretization will help us reduce the dual problem to a finite-dimensional optimization problem. Finally, we will solve the finite-dimensional problem by the use of techniques of convex analysis.

Before proceeding, we remark that the natural generalization of the above theorem to an arbitrary $n$-tuple of real-valued measures is valid. See Theorem 17 below.

2.3 Dual problem

Let us denote by $Y$ the space ${\mathbb{C}}^3/\mathbb{C}\mathbf{1}$ equipped with the norm in (17). Then the dual space $Y^{\ast }$ is the two-dimensional space of three-tuples $(\alpha ,\beta ,\gamma )$ of complex numbers that satisfy

and the norm on $Y^{\ast }$ is given by

In the case $\mathcal{N}=C(X)$⁠, the dual space $(C(X){)}^{\ast }$ is just the space of finite Borel measures on $X$⁠. The adjoint operator ${\mathcal{L}}^{\ast }:Y^{\ast }\to {\mathcal{N}}^{\ast }$ is then given by

and the estimate (20) is equivalent to

Since $\alpha +\beta =-\gamma$ and $(\alpha +\beta +\gamma ){\mu }_3=0$⁠, we may rewrite the above inequality into

where

Note that ${\nu }_1$ and ${\nu }_2$ are real-valued if ${\mu }_1,{\mu }_2,{\mu }_3$ are real-valued. Theorem 7 is thus a consequence of the following slightly more general statement in which the topological structure of $X$ does not play a role.

 

In our next step, we shall simplify the problem further, and show that Proposition 8 can be established by considering finite sets $X$ only.

2.4 Discretization

With notations as in Proposition 8, set $\sigma :=|{\nu }_1|+|{\nu }_2|$⁠. Then $\sigma$ is a positive finite measure on $X$⁠, and by the Radon–Nikodym theorem we have $d{\nu }_1=fd\sigma$ and $d{\nu }_2=gd\sigma$⁠, where $f,g$ are bounded real measurable functions on $X$⁠. For a moment, let $\Vert \cdot {\Vert }_{\sigma ,1}$ denote the norm

Then Proposition 8 is equivalent to the inequality

We will say that a function is simple if it only takes on a finite number of distinct values. By standard measure theory, there exist simple measurable real functions $f_m$⁠, $g_m$ on $X$ such that $f_m\to f$ and $g_m\to g$ uniformly on $X$⁠. Clearly $\Vert \alpha f_m+\beta g_m{\Vert }_{\sigma ,1}\to \Vert \alpha f+\beta g{\Vert }_{\sigma ,1}$⁠. Likewise $\Vert f_m{\Vert }_{\sigma ,1}\to \Vert f{\Vert }_{\sigma ,1}$ and $\Vert g_m{\Vert }_{\sigma ,1}\to \Vert g{\Vert }_{\sigma ,1}$ and $\Vert f_m-g_m{\Vert }_{\sigma ,1}\to \Vert f-g{\Vert }_{\sigma ,1}$⁠. Thus, if the inequality (24) holds for each pair of simple functions, then it holds for $f,g$⁠. So it suffices to establish (24) when $f,g$ are simple measurable real functions.

Hence, suppose that $f,g$ are simple measurable real functions on $X$⁠. We can write them as $f=\sum _{j=1}^n{a}_j{1}_{X_j}$ and $g=\sum _{j=1}^n{b}_j{1}_{X_j}$⁠, where $\{X_1,\dots ,X_n\}$ is a measurable partition of $X$⁠, and $a_j,b_j\in \mathbb{R}$ for all $j$⁠. The inequality in (24) becomes

Writing

and

we see that this becomes

where now $\mathbf{x},\mathbf{y}$ are vectors in ${\mathbb{R}}^n$ and $\Vert \cdot {\Vert }_1$ denotes the usual ${\ell }^1$-norm on ${\mathbb{R}}^n$ given by

To summarize, to prove Proposition 8 and consequently to prove Theorem 7, it suffices to establish the following discrete result.

 

This reduction of the problem to the finite-dimensional setting allows us to use the tools of convex analysis.

2.5 Optimization over a convex set

Consider the set

Thus $C_n$ is a compact convex polytope in ${\mathbb{R}}^n\times {\mathbb{R}}^n$⁠, and so it has a finite number of extreme points. That is, points of $C_n$ that do not lie in the interior of any line segment in $C_n$⁠. A well-known theorem of Carathéodory says that each point of a compact convex polytope is a convex combination of its extreme points.

 

 

From the above lemma and our sequence of reductions above, it follows that in order to prove Theorem 7 it suffices show that the inequality (26) holds at every extreme point of the polytope $C_n$⁠. Proposition 11 below characterizes these extreme points by partitioning them into three equivalence classes.

Note that $C_n$ is invariant under the following linear symmetries:

where $\pi$ is any permutation of $\{1,2,\dots ,n\}$⁠,

for any choice of ${ϵ}_1,\dots ,{ϵ}_n\in \{-1,1\}$⁠,

and

Denote by $G_n$ the group generated by these symmetries. As these symmetries are linear automorphisms of ${\mathbb{R}}^n\times {\mathbb{R}}^n$⁠, it is clear that $G_n$ leaves invariant the set of extreme points $C_n$⁠. We say that two extreme points of $C_n$ are ${\mathit{\text{G}}}_n$-equivalent if there is an element of $G_n$ mapping one of them to the other. Thus the action of $G_n$ on $C_n$ partitions the set of extreme points of $C_n$ into a finite number of equivalence classes. Note that if the inequality (26) holds for some $(\mathbf{x},\mathbf{y})\in C_n$⁠, then it holds also for any point of $C_n$ in the orbit of $(\mathbf{x},\mathbf{y})$ under the group action of $G_n$ on $C_n$⁠.

The extreme points of $C_n$ are identified in the following proposition.

 

One can readily check that each of the three above pairs really is an extreme point of $C_n$⁠. We omit the proof, since we do not actually need this fact. In the case that $n=1$⁠, the same result holds, but only the first kind of pair can arise. Likewise, if $n=2$⁠, the same result holds, but only the first two types of pairs can arise.

We will prove Proposition 11 in Section 2.6. For now let us see how Theorem 7 follows. In order to verify (26) for all extreme points of $C_n$⁠, it suffices to verify the inequality for the three pairs of vectors appearing in Proposition 11. This is an easy task. For instance, if $(\mathbf{x},\mathbf{y})$ is the second pair in Proposition 11, then we have

The inequality for the other two pairs is verified similarly. Then from Lemma 10 we conclude that Proposition 9 holds, from which Theorem 7 follows by the earlier reduction.

It remains to prove Proposition 11.

2.6 Extreme points of the polytope

In the proof of Proposition 11, the group $G_n$ generated by the symmetries (29)–(32) will be extensively used. In particular we will use the property that $(\mathbf{x},\mathbf{y})$ is an extreme point of $C_n$ if and only if some extreme point of $C_n$ is $G_n$-equivalent to it. Moreover, the following two observations will be useful to single out.

 

 

 

 

We are ready to prove Proposition 11. We denote by ${\ell }_n^1$ the space ${\mathbb{R}}^n$ equipped with the norm $\Vert \cdot {\Vert }_1$ given by (25). Recall that the extreme points of the unit ball $B:=\{\mathbf{x}\in {\mathbb{R}}^n:\Vert \mathbf{x}{\Vert }_1\le 1\}$ are the vectors with precisely one non-zero coordinate, this coordinate being equal to $\pm 1$⁠.

 

3 Proof of Theorem 1

In addition to Theorem 7 from Section 2, we will also need some facts from plane geometry in order to prove Theorem 1. In particular, we will need to discuss the minimum enclosing disk problem appearing in computational geometry.

3.1 Minimal enclosing disk

Let $K$ be a compact subset of $\mathbb{C}$ containing at least two points. Among all closed disks that contain $K$ there exists a unique one of minimal radius. We will denote this disk by ${\mathbb{D}}_K$ and call it the minimal disk for $K$⁠. The radius of ${\mathbb{D}}_K$ will be denoted by $R(K)$⁠.

If ${\mathbb{D}}_K$ is minimal for $K$⁠, then the intersection $K\cap \partial {\mathbb{D}}_K$ must obviously be non-empty. In fact, this intersection must contain at least two points, and there is also a restriction on the locations of the points in $K\cap \partial {\mathbb{D}}_K$⁠.

 

 

 

 

3.2 Reduction to three-point sets

The following simple result on minimal disks makes it possible to apply Theorem 7 to more than three measures.

 

It may be convenient to refer to Figure 4 during the reading of the proof.

 

3.3 Finalizing the proof

We are finally ready to give a proof of the equality $c_{\mathbb{R}}(\mathrm{\Omega })=c_{\mathbb{C}}(\mathrm{\Omega })$⁠.

 

The earlier mentioned extension of Theorem 7 to an n-measures theorem is obtained by employing the same argument as in the above proof. The normed space ${\mathbb{C}}^n/\mathbb{C}\mathbf{1}$ appearing below is defined analogously to the case $n=3$ treated in Section 2.1.

 

 

4 Proof of Theorem 2

4.1 Exploiting subsequences

We will argue by contradiction in order to prove Theorem 2. That is, we will assume that there exists a convex domain $\mathrm{\Omega }$ with $a(\mathrm{\Omega })=1$⁠, and so that there exists a sequence of functions $(f_n)$ in $\mathcal{A}(\mathrm{\Omega })$⁠, which satisfy

and

We shall see that this leads to a contradiction. The proof technique below is different from the one employed by Schober in [17] in his proof of Neumann’s lemma, and analyticity is used only at the very end of the proof. In fact, we shall remark at the end of the section how our arguments lead to a new proof of Neumann’s lemma that is different from the one in [17].

Thus, for now, we assume merely that $f_n\in C(\partial \mathrm{\Omega })$⁠, and we will derive certain consequences of (33). In the course of the proof we shall replace the sequence $(f_n)$ by a subsequence multiple times, and for convenience we will not be changing the subscripts. We may suppose that $\Vert f_n{\Vert }_{\mathrm{\Omega }}=1$⁠, and consequently that the images

are contained in a closed disk of radius $1$ centred at the origin. For large $n$⁠, this observation and (33) forces there to be points of the image of $K_{\mathrm{\Omega }}{f}_n$ outside of any disk centred at the origin of radius strictly less than $1$⁠. By exchanging $f_n$ for a unimodular multiple of itself, we may thus assume that there exists a sequence of points $({\zeta }_n)$ in $\partial \mathrm{\Omega }$ for which we have

Using that the functions $f_n$ are bounded by $1$ in modulus, and the positive measures $d{\mu }_{\zeta }$ are of unit mass, we obtain

Recall from (3) that ${\rho }_{{\zeta }_n}$ denotes the $ds$-absolutely continuous part of ${\mu }_{{\zeta }_n}$⁠. The above computation implies that

Compactness of the boundary $\partial \mathrm{\Omega }$ implies that we may assume convergence of the sequence $({\zeta }_n)$ to some points $\zeta \in \partial \mathrm{\Omega }$⁠. The following lemma shows that we may replace in (35) the densities ${\rho }_{{\zeta }_n}$ with the density ${\rho }_{\zeta }$⁠.

 

 

Out next observation extracts more information from (33). Consider the strips

These strips have a fixed large “length” but shrinking “width”. One such strip is marked in Figure 5. We claim that each one of the strips $S_{\delta }$ intersects the images $K_{\mathrm{\Omega }}{f}_n(\mathrm{\Omega })$ non-trivially for infinitely many indices $n$⁠. For if not, then for some fixed $\delta >0$⁠, we would have that $S_{\delta }\cap K_{\mathrm{\Omega }}{f}_n(\partial \mathrm{\Omega })=\mathrm{\varnothing }$ for all sufficiently large $n$⁠, which means that the images $K_{\mathrm{\Omega }}{f}_n(\partial \mathrm{\Omega })$ are entirely contained in $B(0,1)\setminus S_{\delta }$⁠, where $B(0,1)$ denotes the closed disk of radius $1$ centred at the origin. But if ${ϵ}_1$ and ${ϵ}_2$ are sufficiently small positive numbers, then $B(0,1)\setminus S_{\delta }\subset B({ϵ}_1,1-{ϵ}_2)$⁠, a disk of radius $1-{ϵ}_2$ centred at the point ${ϵ}_1\in \mathbb{R}$⁠. See Figure 5. Recalling the geometric interpretation of the norm $\Vert K_{\mathrm{\Omega }}{f}_n+\mathbb{C}\mathbf{1}{\Vert }_{\partial \mathrm{\Omega }}$ as the radius of the smallest disk containing the image of $K_{\mathrm{\Omega }}{f}_n$⁠, we would arrive at a contradiction to (33). Thus every strip $S_{\delta }$ contains points in the image of $K_{\mathrm{\Omega }}{f}_n$ for infinitely many $n$⁠.

 

 

4.2 Proof of Theorem 2

The above arguments are valid for $f_n\in C(\partial \mathrm{\Omega })$⁠. However, under the assumption of analyticity, the sequence $(f_n)$ cannot converge to two different constants on two different sets of positive arclength measure. To make this statement precise, we appeal to the classical theory of analytic functions in the (open) unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$⁠. Here [8, Chapter II] is an excellent reference for the claims made in the following proof.

 

4.3 A proof of Neumann’s lemma

We indicate how one may proceed to use our above arguments to obtain a proof of Neumann’s lemma, stating that $c(\mathrm{\Omega })=1$ if and only if $\mathrm{\Omega }$ is a triangle or a quadrilateral. We need only the following simple geometric observation regarding the densities ${\rho }_{\zeta }$⁠.

 

 

Neumann’s lemma is established as follows. Assume that $c(\mathrm{\Omega })=1$⁠. From Lemmas 18 and 19 we see that two points $\zeta ,{\zeta }^{\prime }$ exist for which the measures ${\rho }_{\zeta }ds$ and ${\rho }_{{\zeta }^{\prime }}ds$ are mutually singular. From Lemma 20 we deduce that the support of ${\rho }_{\zeta }ds$ is the union of at most two line segments containing ${\zeta }^{\prime }$⁠, and the complement of the support of ${\rho }_{\zeta }ds$ is also a union of at most two line segments. Thus $\partial \mathrm{\Omega }$ is the union of at most four line segments.

5 Examples

In this section, we compute and estimate the configuration constants for some types of domains.

5.1 Configuration constant of an ellipse

For $a,b>0$⁠, let

be the ellipse centred at the origin with semi-axes of lengths $a$ and $b$⁠, respectively. It is quite remarkable that the configuration constant can in this case be computed explicitly.

 

In order to prove the proposition, our first step is to derive an expression for the density of the Neumann–Poincaré kernel of ${\mathrm{\Omega }}_{a,b}$⁠. The boundary $\partial {\mathrm{\Omega }}_{a,b}$ is parametrized by

Here $[0,2\pi ]$ can be replaced by any interval of length $2\pi$⁠. Recalling formula (4) for ${\mu }_{\zeta }$ and setting $\zeta =\gamma (s)$⁠, $\sigma =\gamma (t)$⁠, we obtain

Using (37), this formula can be greatly simplified.

 

The lemma is established by combining (37) and (38), and then using elementary trigonometric identities to simplify the resulting expression.

With this formula in hand, we now evaluate the configuration constant of the ellipse ${\mathrm{\Omega }}_{a,b}$⁠.

 

5.2 Integral estimates

For a general domain, the exact value of $c(\mathrm{\Omega })$ is often inaccessible. In this section, we will present a simple estimate that is applicable to domains $\mathrm{\Omega }$ with a non-flat part of the boundary that leads to an upper bound on $c(\mathrm{\Omega })$⁠.

Assume that we find a Borel measure $\nu$ on $\partial \mathrm{\Omega }$ such that

If so, then, for every $\varphi \in C(\partial \mathrm{\Omega })$ with $\Vert \varphi {\Vert }_{\partial \mathrm{\Omega }}\le 1$⁠, we have

which shows that the image of $K_{\mathrm{\Omega }}\varphi$ is contained in a disk of radius $k_{\mathrm{\Omega }}^{\nu }$ centred at ${\int }_{\partial \mathrm{\Omega }}\varphi \text{d}\nu$⁠. Thus,

One approach is to seek a positive measure $\nu$ on $\partial \mathrm{\Omega }$ satisfying ${\mu }_{\zeta }\ge \nu$ for all $\zeta \in \partial \mathrm{\Omega }$⁠. Then

and so $k_{\mathrm{\Omega }}^{\nu }=1-\nu (\partial \mathrm{\Omega })$⁠.

We will construct the largest non-negative Borel measure $\nu$ on $\partial \mathrm{\Omega }$⁠, which satisfies ${\mu }_{\zeta }-\nu \ge 0$⁠. The construction is based on the geometric interpretation of the density ${\rho }_{\zeta }(\sigma )$ in (5) and the quantity $R_{\mathrm{\Omega }}$ appearing in (15). In order to avoid the need to establish Borel measurability of $R_{\mathrm{\Omega }}$ defined as a supremum of an uncountable family as in (15), we proceed to define it in a slightly different but equivalent way. Namely, it is easy to see that, given $\sigma \in \partial \mathrm{\Omega }$⁠, if there exists a closed disk $\mathrm{\Delta }$ such that $\mathrm{\Omega }\subset \mathrm{\Delta }$ and $\sigma \in \partial \mathrm{\Delta }$⁠, then there exists one of smallest radius. We denote this radius by $R_{\mathrm{\Omega }}(\sigma )$⁠. Note that if $\sigma$ is not a corner of $\partial \mathrm{\Omega }$⁠, then the corresponding disk must be tangent to $\partial \mathrm{\Omega }$ at $\sigma$⁠. If no disk passing through $\sigma$ exists that contains $\mathrm{\Omega }$⁠, then we set $R_{\mathrm{\Omega }}(\sigma ):=\mathrm{\infty }$⁠. This happens, for instance, if $\sigma$ is contained in the interior of a line segment in $\partial \mathrm{\Omega }$⁠. In particular, $R_{\mathrm{\Omega }}(\sigma )=\mathrm{\infty }$ for all but a finite number of points of any polygonal domain.

 

 

We set

By the above lemma, $\nu$ is a non-negative Borel measure on $\partial \mathrm{\Omega }$⁠. For any $\zeta \in \partial \mathrm{\Omega }$⁠, we have

To see this, note that if $\sigma$ is not a corner and $R_{\zeta ,\sigma }$ is the radius of the unique circle tangent to $\partial \mathrm{\Omega }$ at $\sigma$ and passing through $\zeta$⁠, then $R_{\mathrm{\Omega }}(\sigma )\ge R_{\zeta ,\sigma }$⁠. Therefore, according to (5),

for almost every $\sigma$ with respect to arclength measure on $\partial \mathrm{\Omega }$⁠. Inequality (41) follows. Although we shall skip a formal proof, we mention also that $\nu$ is in fact the largest measure satisfying ${\mu }_{\zeta }\ge \nu$ for all $\zeta \in \partial \mathrm{\Omega }$⁠. This maximality property of $\nu$ is to be interpreted in the following sense: if ${\nu }^{\prime }$ is any measure satisfying ${\mu }_{\zeta }\ge {\nu }^{\prime }$ for all $\zeta$⁠, then $\nu \ge {\nu }^{\prime }$⁠.

By our earlier discussion, we obtain the following upper estimate for the configuration constant:

Note that this is precisely the assertion of Theorem 6 stated in Section 1.

We will now mention some consequences. Recall that if $\gamma$ is a plane curve of class $C^2$⁠, then the radius of curvature of $\gamma$ is the reciprocal of its curvature.

 

 

This last result was already known. See for example [7, pp. 45–46] and [12, pp. 128–129]. However the proofs in these references are quite different from the one above.

 

 

5.3 Analytic configuration constants of quadrilaterals

Theorem 2 shows that $a(\mathrm{\Omega })<1$ for every $\mathrm{\Omega }$⁠. Here we show by example that $a(\mathrm{\Omega })$ may be arbitrarily close to $1$⁠. Figure 7 shows a narrow quadrilateral domain for which this phenomenon occurs.

 

 

5.4 Configuration constants equal to zero

Recall the estimate (13) from Section 1, which will be proved in the next section. The estimate is strongest if $a(W)=0$⁠, and in this case we reach the conjectured bound $K=2$⁠. Unfortunately, the only domain $W$ for which we have $a(W)=0$ is a disk, and in this case (13) reduces to the well-known Okubo–Ando bound from [15]. For completeness we give a proof of the statement that $a(\mathrm{\Omega })=0$ if and only if $\mathrm{\Omega }$ is a disk. More precisely, we have the following.

 

 

6 Application to Numerical Ranges

6.1 Spectral constant estimate

Our principal motivation for the introduction of the analytic configuration constant is the following result that was mentioned in the Section 1 and that we will now prove.

 

Of course, if $W$ has no interior, then its convexity forces it to be a line segment. In that case $T$ is a normal operator, and the spectral theorem gives us the better estimate $\Vert f(T)\Vert \le \Vert f{\Vert }_{\sigma (T)}$⁠, where $f$ may be any Borel measurable function on the spectrum $\sigma (T)$⁠. Thus Theorem 28 implies Theorem 3. In what follows, we will assume that $W$ has non-empty interior.

Let us make some initial remarks before going into the proof of Theorem 28. In the case $\sigma (T)$ is contained in the interior of $W$⁠, then $f(T)$ is defined, as usual, through the Dunford–Riesz holomorphic functional calculus. If $\partial W\cap \sigma (T)\ne \mathrm{\varnothing }$⁠, then this definition does not work. Nevertheless, if $f\in \mathcal{A}(W)$⁠, then it is a standard result of approximation theory that a sequence of analytic polynomials $(p_n)$ exists, which converges to $f$ uniformly on $W$⁠. In the presence of any uniform bound of the form $\Vert p(T)\Vert \le K\Vert p(T){\Vert }_W$ for polynomials $p$⁠, we may then define $f(T)$ as the limit of the sequence $(p_n(T))$ in the operator norm. Such bounds are known to exists, the strongest known bound $K\le 1+\sqrt{2}$ being due to Crouzeix and Palencia. Theorem 28 improves this estimate given information about the numerical range of $T$⁠.

Our proof of Theorem 28 combines the argument of Crouzeix and Palencia from [5] with ideas of Schwenninger and de Vries from [18], where bounds for various functional calculi are derived as a consequence of the existence of extremal functions and extremal vectors. Let $U$ be an open set in the plane, and $H^{\mathrm{\infty }}(U)$ be the algebra of bounded holomorphic functions on $U$⁠. Given an operator $T:\mathcal{H}\to \mathcal{H}$ with $\sigma (T)$ contained in $U$⁠, it is elementary that the quantity

is finite. A normal-families argument shows that an $f\in H^{\mathrm{\infty }}(U)$ exists with $\Vert f{\Vert }_U=1$ for which the supremum above is attained. Any such $f$ will be called for an extremal function. If, moreover, a vector $x\in \mathcal{H}$ with $\Vert x{\Vert }_{\mathcal{H}}=1$ exists for which

then we will say that $x$ is an extremal vector, and $(f,x)$ is an extremal pair. Unless $\dim \mathcal{H}<\mathrm{\infty }$⁠, an extremal vector might not exist, but we will be able to reduce the proof to the finite-dimensional case. The importance of the concept of extremal pairs $(f,x)$ stems from the following result. We refer the reader to [1, Theorem 4.5] for a proof (see also [18, Proposition 3]).

 

The next two lemmas will reduce our task to consideration of finite-dimensional Hilbert spaces, in which extremal vectors exist, and will dispose of the problematic set $\sigma (T)\cap \partial W$⁠. The first observation is essentially contained in [18, Proposition 9].

 

 

The proof of the next lemma will use affine invariance of the configuration constants. Let us fix $\alpha ,\beta \in \mathbb{C}$⁠, $\alpha \ne 0$⁠, and an affine mapping $A(z):=\alpha z+\beta$⁠. Then $A$ is a conformal transformation of $\mathbb{C}$ with the additional property of taking a line segment of length $L$ to a line segment of length $|\alpha |L$⁠, and a circle of radius $R$ to a circle of radius $|\alpha |R$⁠. Let $\tilde{\mathrm{\Omega }}=A(\mathrm{\Omega })$ be the affine image of $\mathrm{\Omega }$ under $A$⁠, and recall the formula for the Neumann–Poincaré kernel in (3) and its geometric interpretation. If $\zeta ,\sigma \in \partial \mathrm{\Omega }$⁠, $E$ is a Borel subset of $\partial \mathrm{\Omega }$⁠, and $s,\tilde{s}$ are the arclength measures on $\partial \mathrm{\Omega }$ and $\partial \tilde{\mathrm{\Omega }}$⁠, respectively, then it follows from the properties of $A$ listed above that

• (i)

  ${\theta }_{\zeta }={\theta }_{A(\zeta )}$⁠,

• (ii)

  $|\alpha |s(E)=\tilde{s}(A(E))$⁠,

• (iii)

  $|\alpha |R_{\zeta ,\sigma }=R_{A(\zeta ),A(\sigma )}$⁠.

A consequence is that the Neumann–Poincaré kernels $\{{\mu }_{\zeta }{\}}_{\zeta \in \partial \mathrm{\Omega }}$ and $\{{\tilde{\mu }}_{A(\zeta )}{\}}_{A(\zeta )\in \partial \mathrm{\Omega }}$ of the respective domains satisfy

Then a change of variables shows that $K_{\mathrm{\Omega }}(\tilde{f}\circ A)=K_{\tilde{\mathrm{\Omega }}}\tilde{f}$ for any $\tilde{f}\in C(\partial \tilde{\mathrm{\Omega }})$⁠, and it follows that

Armed with these equalities, we make our second observation.

 

 

 

Funding

This work has been done during Malman’s visit at Département de mathématiques et de statistique, Université Laval, supported by Simons-CRM Scholar-in-Residence program. Mashreghi’s research was supported by an NSERC Discovery Grant and the Canada Research Chairs program. O’Loughlin was supported by a CRM-Laval Postdoctoral Fellowship. Ransford’s research was supported by an NSERC Discovery Grant.

Acknowledgments

We are grateful to the referee for the careful reading of the paper, and for the suggestions that we used to improve the manuscript.

A Double-Layer Potential on a General Convex Domain

A.1 Convex domains

Let $\mathrm{\Omega }$ be a compact convex domain in the plane $\mathbb{C}$ with non-empty interior ${\mathrm{\Omega }}^o$⁠. We will be making no assumptions regarding smoothness of the boundary $\partial \mathrm{\Omega }$⁠. However, convexity itself implies that $\partial \mathrm{\Omega }$ is a rectifiable simple closed curve with some additional properties.

The orientation of $\partial \mathrm{\Omega }$ is to be counter-clockwise (i.e., positive), and we use ${\sigma }^{\prime }\uparrow \sigma$ and ${\sigma }^{\prime }\downarrow \sigma$ to denote, respectively, the counter-clockwise and clockwise one-sided convergence of ${\sigma }^{\prime }$ to $\sigma$ within $\partial \mathrm{\Omega }$⁠. As a consequence of convexity of $\mathrm{\Omega }$⁠, the one-sided tangent angles exist at every point $\sigma \in \partial \mathrm{\Omega }$⁠, are locally given by

and satisfy

Strict inequality may occur at most at a countable subset of $\partial \mathrm{\Omega }$⁠. If it occurs at $\sigma$⁠, then we say that $\partial \mathrm{\Omega }$ has a corner at $\sigma$⁠. At any point that is not a corner, the tangent angle

is well-defined, and so is the tangent $T(\sigma ):=e^{i\alpha (\sigma )}$ itself. If $t\mapsto \gamma (t)$ is any (positively-oriented) parametrization of $\partial \mathrm{\Omega }$⁠, and we set $\alpha (\sigma )={\alpha }_{+}(\sigma )$ at the corners, then the locally defined function $\alpha (\gamma (t))$ is increasing in $t$⁠, and consequently the tangent $T$ is continuous at every point that is not a corner of $\partial \mathrm{\Omega }$⁠. At a corner, the discontinuity of $T$ amounts to a jump of the argument of $T$⁠. We denote by $N(\sigma ):=-iT(\sigma )$ the outward-pointing normal at $\sigma \in \partial \mathrm{\Omega }$⁠.

A.2 Double-layer potential

Let ${\mathrm{\Omega }}^o$ denote the interior of $\mathrm{\Omega }$⁠. To each $z\in {\mathrm{\Omega }}^o$ we associate the measure ${\mu }_z$ on $\partial \mathrm{\Omega }$⁠, which for any arc $J\subset \partial \mathrm{\Omega }$ satisfies

Here $\arg (\sigma -z)$ is any locally defined continuous determination of the argument function on $\partial \mathrm{\Omega }$⁠. Non-negativity of ${\mu }_z$ follows from convexity of $\mathrm{\Omega }$ and our choice of positive orientation of $\partial \mathrm{\Omega }$⁠. With respect to this orientation, every arc $J=(a,b)\subset \partial \mathrm{\Omega }$ has a start-point $a$ and an end-point $b$⁠, and it is easy to see that

In particular, ${\mu }_z(\partial \mathrm{\Omega })=2$⁠.

The measure ${\mu }_z$ is absolutely continuous with respect to arclength $s$ on $\partial \mathrm{\Omega }$⁠. Indeed, if ${\sigma }_0\in \partial \mathrm{\Omega }$⁠, $J_n=(a_n,b_n)$ is a sequence of arcs of $\partial \mathrm{\Omega }$ that are shrinking to ${\sigma }_0$⁠, and $|J_n|$ are the corresponding arclengths, then

We use above an appropriate locally defined holomorphic branch of the logarithm. As $n\to \mathrm{\infty }$⁠, the first factor inside the brackets satisfies

while the second factor stays bounded as a consequence of the inequality $|b_n-a_n|\le |J_n|$⁠. Thus

and from elementary measure theory we obtain that ${\mu }_z$ is absolutely continuous with respect to $s$⁠. If moreover ${\sigma }_0$ is not a corner, then it can be shown that

and so in additional to boundedness we even have the convergence

Thus the Radon–Nikodym derivative satisfies

at every $\sigma \in \partial \mathrm{\Omega }$⁠, which is not a corner.

A.3 Boundary kernel

The Neumann–Poincaré kernel is the boundary version of the family of measures $\{{\mu }_z{\}}_{z\in {\mathrm{\Omega }}^o}$ introduced above. To each point $\zeta \in \partial \mathrm{\Omega }$ we associate the Borel probability measure on $\partial \mathrm{\Omega }$ defined by (A.1) for arcs $J\subset \partial \mathrm{\Omega }$ not containing the point $\zeta$⁠. Because $\zeta \in \partial \mathrm{\Omega }$⁠, this definition implies that ${\mu }_{\zeta }(\partial \mathrm{\Omega }\setminus \{\zeta \})={\theta }_{\zeta }/\pi$⁠, where ${\theta }_{\zeta }=\pi -{\alpha }_{+}(\zeta )+{\alpha }_{-}(\zeta )$ can be interpreted as the angle of the aperture at $\zeta$⁠. Indeed, ${\theta }_{\zeta }$ is equal to the increase in the argument of $\sigma -\zeta$ as we traverse one loop around $\partial \mathrm{\Omega }$ starting and ending at the point $\zeta$⁠, and since ${\mu }_{\zeta }$ is a probability measure, we must have

With the exception of this possible point mass, ${\mu }_{\zeta }$ is otherwise absolutely continuous with respect to arclength. The corresponding Radon–Nikodym derivative is given by

The formula (A.3) is established analogously to (A.2). All in all, the measure ${\mu }_{\zeta }$ can be decomposed as

where ${\delta }_{\zeta }$ is a unit mass at $\zeta \in \partial \mathrm{\Omega }$⁠, ${\theta }_{\zeta }$ is the angle of the aperture at $\zeta$ (with the convention that ${\theta }_{\zeta }=\pi$ if $\zeta$ is not a corner), and where the density ${\rho }_{\zeta }$ is given by (A.3).

A.4 Weak-star convergence

We establish now that

in the sense of the weak-star topology on measures. Note that if $B=B(\zeta ,\delta )$ is a ball of radius $\delta >0$ centred at $\zeta \in \partial \mathrm{\Omega }$⁠, then expressions (A.2) and (A.3) for the densities of ${\mu }_z$ and ${\mu }_{\zeta }$ show that

for every $f\in C(\partial \mathrm{\Omega })$⁠. In particular, choosing $f=\mathbf{1}$⁠, we obtain

Since

we see that given $ϵ>0$ for all sufficiently small $\delta >0$ we will have

Returning to general $f\in C(\partial \mathrm{\Omega })$⁠, we have

On the right-hand side, the first term tends to zero as $z\to \zeta$⁠, the second can be made arbitrarily small by continuity of $f$⁠, the crude estimate ${\mu }_z(B)\le 2$ and choice of sufficiently small $\delta$⁠, the third is dominated in modulus by $\Vert f{\Vert }_{\partial \mathrm{\Omega }}\cdot ϵ$ for $z$ sufficiently close to $\zeta$⁠, and the fourth is dominated by $\Vert f{\Vert }_{\partial \mathrm{\Omega }}\cdot {\mu }_{\zeta }(B\setminus \{\zeta \})$⁠, which also can be made arbitrarily small by choice of sufficiently small $\delta$⁠. The desired weak-star convergence follows.

References