A discrete version of Koldobsky's slicing inequality

2015/11/09 by Alexander, Matthew, Henk, Martin, Zvavitch, Artem · 1 citation
#52A20 #52B10 #53A15 #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1511.02702

Abstract

Let # K be a number of integer lattice points contained in a set K. In this paper we prove that for each d∈ \mathbb N there exists a constant C(d) depending on d only, such that for any origin-symmetric convex body K ⊂ \mathbb R^d containing d linearly independent lattice points # K ≤ C(d)max(# (K∩ H)) \rm vol_d(K)^(d-m)/(d), where the maximum is taken over all m-dimensional subspaces of \mathbb R^d. We also prove that C(d) can be chosen asymptotically of order O(1)^dd^d-m. In addition, we show that if K is an unconditional convex body then C(d) can be chosen asymptotically of order O(d)^d-m.

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• From Brunn-Minkowski to Prékopa-Leindler and Borell-Brascamp-Lieb: discrete inequalities

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