报告人：Dylan Langharst（法国巴黎左岸数学研究所）

邀请人：张宁

报告时间：2024年5月8日（星期三）14:30-16:30

报告地点：东三十一楼119室

报告题目：Weighted Minkowski's Existence Theorem

报告摘要：Minkowski once considered: what are the necessary and sucient conditions for a collection of unit vectors fuig and positive numbers faig to correspond to the convex polytope whose ith face has outer-unit normal ui and surface area ai? His famed existence theorem found there are merely two minor conditions needed to guarantee the existence of a unique, centered polytope with the prescribed properties.For an arbitrary convex body, whose resolution follows from the polytope case, one shows a Borel measure.on the unit sphere is the surface area measure (SK) for a unique centered convex body K.In this talk, we consider replacing volume with some Borel measure that has density. We consider a rich class of Borel measures and solve the following weighted Minkowski's existence theorem: for a nite,even Borel measure on the unit sphere and  an even Borel measure on Rn from the rich class, there exists a symmetric convex body K in Rn such that d(u) = c;KdS;K(u);where c;K is a quantity that depends on  and K and dS;K(u) is the weighted surface area-measure of K with respect to. Examples of measures in the rich class are homogeneous measures (with c;K = 1) and radially decreasing probability measures with continuous densities (e.g. the Gaussian measure). Under certain concavity conditions on the measure, we also obtain uniqueness.Joint with L. Kryvonos.

报告人简介：Dylan Langharst，法国巴黎左岸数学研究所博士后，肯特州立大学博士，师从Artem Zvavitch，主要研究凸几何分析、几何概率等方向，现在Trans of AMS、Proc of LMS、IMRN等优秀期刊上发表文章。