On discrete L_p Brunn–Minkowski type inequalities

Abstract

L_p Brunn–Minkowski type inequalities for the lattice point enumerator mathrm {G}_n(cdot ) are shown, pge 1, both in a geometrical and in a functional setting. In particular, we prove that Gn((1-λ)·K+pλ·L+(-1,1)n)p/n≥(1-λ)Gn(K)p/n+λGn(L)p/n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned}\\mathrm {G}_n\\bigl ((1-\\lambda )\\cdot K +_p \\lambda \\cdot L + (-1,1)^n\\bigr )^{p/n}\\ge (1-\\lambda )\\mathrm {G}_n(K)^{p/n}+\\lambda \\mathrm {G}_n(L)^{p/n} \\end{aligned}$$\\end{document}for any K, Lsubset mathbb {R}^n bounded sets with integer points and all lambda in (0,1). We also show that these new discrete analogues (for mathrm {G}_n(cdot )) imply the corresponding results concerning the Lebesgue measure.