Variants of the Busemann-Petty problem and of the Shephard problem

2016/01/10 by Apostolos Giannopoulos, Giannopoulos, Apostolos, Alexander Koldobsky +1
Mathematics · #52A20 #FOS: Mathematics #Metric Geometry (math.MG) #math.MG #msc:52A20

paper · pdf · doi:10.48550/arxiv.1601.02231

arxiv created 2016/01/16 · arxiv updated 2016/01/19

Abstract

We provide an affirmative answer to a variant of the Busemann-Petty problem, proposed by V.~Milman: Let K be a convex body in \mathbb Rⁿ and let D be a compact subset of \mathbb Rⁿ such that, for some 1\ls k\ls n-1, |P_F(K)|\ls |D∩ F| for all F∈ G_n,k, where P_F(K) is the orthogonal projection of K onto F and D∩ F is the intersection of D with F. Then, |K|\ls |D|. We also provide estimates for the lower dimensional Busemann-Petty and Shephard problems, and we prove separation in the original Busemann-Petty problem.

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