Inequalities for the quermassintegrals of sections of convex bodies

2023/02/21 by Liakopoulos, Dimitris-Marios
#FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2302.10568

Abstract

We provide general estimates which compare the quermassintegrals of a convex body K in \mathbb Rⁿ with the averages of the corresponding quermassintegrals of the k-codimensional sections of K over G_n,n-k. An example is the inequality α_n,k,j(W_j(K))/(|K|)≤∫_G_n,n-k(W_j(K∩ F))/(|K∩ F|)dν_n,n-k(F)≤ β_n,k,j(W_j(K))/(|K|) where the constants α_n,k,j and β_n,k,j depend only on n,k and j, which holds true for any centrally symmetric convex body K in \mathbb Rⁿ and any 0≤ j≤ n-k-1≤ n-1. Using these estimates we obtain some positive results for suitable versions of the slicing problem for the quermassintegrals of a convex body.

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