2021/11/22 by Alonso-Gutiérrez, David, Lucas, Eduardo, Nicolás, Jesús Yepes · 1 citation
#26D15 #FOS: Mathematics #Metric Geometry (math.MG) #Primary 52C07 #Secondary 52A40
paper · doi:10.48550/arxiv.2111.11533
In this paper we study various Rogers-Shephard type inequalities for the lattice point enumerator Gn(⋅) on ℝn. In particular, for any non-empty convex bounded sets K,L⊂ℝn, we show that Gn(K+L)Gn(K∩(-L)) ≤\binom2nn Gn(K+(-1,1)n)Gn(L+(-1,1)n). and Gn-k(PH^⊥ K)Gk(K∩ H)≤\binomnkGn(K+(-1,1)n), for H=lin\e1,…,ek\, k∈\1,…,n-1\. Additionally, a discrete counterpart to a classical result by Berwald for concave functions, from which other discrete Rogers-Shephard type inequalities may be derived, is shown. Furthermore, we prove that these new discrete analogues for Gn(⋅) imply the corresponding results involving the Lebesgue measure.