On a quantitative reversal of Alexandrov's inequality

2017/02/19 by Paouris, Grigoris, Pivovarov, Peter, Valettas, Petros
#FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1702.05762

Abstract

Alexandrov's inequalities imply that for any convex body A, the sequence of intrinsic volumes V₁(A),…,V_n(A) is non-increasing (when suitably normalized). Milman's random version of Dvoretzky's theorem shows that a large initial segment of this sequence is essentially constant, up to a critical parameter called the Dvoretzky number. We show that this near-constant behavior actually extends further, up to a different parameter associated with A. This yields a new quantitative reverse inequality that sits between the approximate reverse Urysohn inequality, due to Figiel--Tomczak-Jaegermann and Pisier, and the sharp reverse Urysohn inequality for zonoids, due to Hug--Schneider. In fact, we study concentration properties of the volume radius and mean width of random projections of A and show how these lead naturally to such reversals.

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