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P. R. Goodey, Limits of Intermediate Surface Area Measures of Convex Bodies, Proceedings of the London Mathematical Society, Volume s3-43, Issue 1, July 1981, Pages 151–168, https://doi.org/10.1112/plms/s3-43.1.151
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Abstract
Let Ij denote the set of jth-order surface area measures of convex sets in Ed (in the Aleksandrov-Fenchel-Jessen sense). In a recent paper, Wolfgang Weil has shown that if μ ε Ij then dim μ (the dimension of the linear hull of the support of μ) is d, d–j, or zero, and that if dim μ = d–j then μ is essentially (d–j–1)-dimensional spherical Lebesgue measure. He also showed that if d–j+1≤n≤d–1 then there is a sequence (μ1)1∞ = 1 of measures in Ij and a measure μ such that dim μ=n and μi→μ weakly as i → ∞. In the present paper we complement these results by showing that if μ(≠0) is the weak limit of such a sequence then dim μ≥d–j, and that if dim μ=d–j then μ is essentially (d–j–1)-dimensional spherical Lebesgue measure. This verifies two of Weil's conjectures. We conclude the work by showing that if ℳ is the set of all positive Borel measures on Sd-1 which have barycentre 0 then, for j<d-1, Ij is a small subset of ℳ in the Baire category sense. This observation contrasts with Weil's demonstration that Ij–Ij is dense in ℳ–ℳ.
