An extension of Minkowski's theorem and its applications to questions about projections for measures
An extension of Minkowski's theorem and its applications to questions about projections for measures
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DOI：
10.1016/j.aim.2019.106803
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发表时间：
2016-07
影响因子：
1.7
通讯作者：
G. Livshyts
中科院分区：
文献类型：
--
作者：
G. Livshyts
闵可夫斯基定理断言，球面上不集中在一个大子球面上的每一个中心测度都是某个凸体的表面积测度，并且表面积测度唯一地确定一个凸体直到移位。本文证明了闵可夫斯基定理的一个推广。考虑Rn上的测度μ具有正的齐次度和正的齐次度.我们证明了凸集K的表面积测度，关于μ加权，唯一确定凸体直到μ-测度为零。我们还建立了一个存在性结果的自然条件下，包括对称性。我们应用这个结果来推广经典Shephard问题的解。为此，我们引入了一个新的概念，它将凸体的投影与给定的测度μ联系起来，并且是投影的Lebesgue体积的推广。
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 R n 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.