Random ball-polyhedra and inequalities for intrinsic volumes

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• 作者：Grigoris Paouris ; Peter Pivovarov
• 关键词：Convex body ; Mean width ; Minkowski symmetrization ; Steiner symmetrization ; Rearrangement inequalities ; Wulff shape ; Generalized Urysohn inequality ; Intersections of congruent balls
• 刊名：Monatshefte für Mathematik
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：182
• 期：3
• 页码：709-729
• 全文大小：
• 刊物主题：Mathematics, general;
• 出版者：Springer Vienna
• ISSN：1436-5081
• 卷排序：182

文摘

We prove a randomized version of the generalized Urysohn inequality relating mean width to the other intrinsic volumes. To do this, we introduce a stochastic approximation procedure that sees each convex body K as the limit of intersections of Euclidean balls of large radii and centered at randomly chosen points. The proof depends on a new isoperimetric inequality for the intrinsic volumes of such intersections. If the centers are i.i.d. and sampled according to a bounded continuous distribution, then the extremizing measure is uniform on a Euclidean ball. If one additionally assumes that the centers have i.i.d. coordinates, then the uniform measure on a cube is the extremizer. We also discuss connections to a randomized version of the extended isoperimetric inequality and symmetrization techniques.