Brunn-Minkowski type inequalities for the lattice point enumerator

2019/11/28 by Iglesias, David, Nicolás, Jesús Yepes, Zvavitch, Artem · 2 citations
#FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1911.12874

Abstract

Geometric and functional Brunn-Minkowski type inequalities for the lattice point enumerator G_n(⋅) are provided. In particular, we show that G_n((1-λ)K + λL + (-1,1)ⁿ)^1/n≥ (1-λ)G_n(K)^1/n+λG_n(L)^1/n for any non-empty bounded sets K, L⊂ℝⁿ and all λ∈(0,1). We also show that these new discrete versions imply the classical results, and discuss some links with other related inequalities.

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