On discrete $L_p$ Brunn-Minkowski type inequalities

Abstract

$L_p$ Brunn-Minkowski type inequa\-li\-ties for the lattice point enumerator ${\mathrm{G}}_n(\cdot )$ are shown, both in a geometrical and in a functional setting. In particular, we prove that \[\mathrm{G}_n\bigl((1-λ)\cdot K +_p λ\cdot L + (-1,1)^n\bigr)^{p/n}\geq (1-λ)\mathrm{G}_n(K)^{p/n}+λ\mathrm{G}_n(L)^{p/n}\] for any $K,L\subset {\mathbb{R}}^n$ 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.