2004/06/21 by Artem Zvavitch, Zvavitch, Artem · 1 citation
Mathematics · Medicine · #52A15 #52A21 #52A38 #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #Pharmacological Effects of Medicinal Plants #Point processes and geometric inequalities #math.FA #math.MG #msc:52A15 #msc:52A21 #msc:52A38
paper · pdf · doi:10.48550/arxiv.math/0406406
arxiv created 2004/06/21 · openalex publication_date 2004/06/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The aim of this paper is to study properties of sections of convex bodies with respect to different types of measures. We present a formula connecting the Minkowski functional of a convex symmetric body K with the measure of its sections. We apply this formula to study properties of general measures most of which were known before only in the case of the standard Lebesgue measure. We solve an analog of the Busemann-Petty problem for the case of general measures. In addition, we show that there are measures, for which the answer to the generalized Busemann-Petty problem is affirmative in all dimensions. Finally, we apply the latter fact to prove a number of different inequalities concerning the volume of sections of convex symmetric bodies in \Rn and solve a version of generalized Busemann-Petty problem for sections by k-dimensional subspaces.