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- W2974206593 abstract "Minkowski's Theorem asserts that every centered measure on the sphere which is not concentrated on a great subsphere is the surface area measure of some convex body, and the surface area measure determines a convex body uniquely up to a shift. In this manuscript we prove an extension of Minkowski's theorem. Consider a measure μ on Rn with positive degree of concavity and positive degree of homogeneity. We show that a surface area measure of a convex set K, weighted with respect to μ, determines a convex body uniquely up to μ-measure zero. We also establish an existence result under natural conditions including symmetry. We apply this result to extend the solution to classical Shephard's problem. To do this, we introduce a new notion which relates projections of convex bodies to a given measure μ, and is a generalization of the Lebesgue volume of a projection." @default.
- W2974206593 created "2019-09-26" @default.
- W2974206593 creator A5075539741 @default.
- W2974206593 date "2019-11-01" @default.
- W2974206593 modified "2023-10-03" @default.
- W2974206593 title "An extension of Minkowski's theorem and its applications to questions about projections for measures" @default.
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- W2974206593 doi "https://doi.org/10.1016/j.aim.2019.106803" @default.
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