Randomized isoperimetric inequalities

2016/07/02 by Grigoris Paouris, Paouris, Grigoris, Peter Pivovarov +1 · 2 citations
Mathematics · #FOS: Mathematics #Metric Geometry (math.MG) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1607.00519

openalex publication_date 2016/07/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We discuss isoperimetric inequalities for convex sets. These include the classical isoperimetric inequality and that of Brunn-Minkowski, Blaschke-Santalo, Busemann-Petty and their various extensions. We show that many such inequalities admit stronger randomized forms in the following sense: for natural families of associated random convex sets one has stochastic dominance for various functionals such as volume, surface area, mean width and others. By laws of large numbers, these randomized versions recover the classical inequalities. We give an overview of when such stochastic dominance arises and its applications in convex geometry and probability.

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• Fixed and periodic points of the intersection body operators of lower orders
• On stochastic forms of functional isoperimetric inequalities

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