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Limits of Intermediate Surface Area Measures of Convex Bodies
Author(s) -
Goodey P. R.
Publication year - 1981
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-43.1.151
Subject(s) - mathematics , lebesgue measure , combinatorics , measure (data warehouse) , borel measure , borel set , convex hull , sequence (biology) , lebesgue integration , dimension (graph theory) , regular polygon , probability measure , convex body , surface (topology) , null set , limit (mathematics) , complement (music) , convex set , zero (linguistics) , mathematical analysis , geometry , set (abstract data type) , computer science , biology , genetics , biochemistry , programming language , complementation , gene , phenotype , convex optimization , philosophy , database , linguistics , chemistry
Let I j denote the set of jth‐order surface area measures of convex sets in E d (in the Aleksandrov‐Fenchel‐Jessen sense). In a recent paper, Wolfgang Weil has shown that if μ ε I j 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 (μ 1 ) 1 ∞ = 1 of measures in I j and a measure μ such that dim μ=n and μ i →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 S d‐1 which have barycentre 0 then, for j < d ‐1, I j is a small subset of ℳ in the Baire category sense. This observation contrasts with Weil's demonstration that I j – I j is dense in ℳ–ℳ.