On discrete $$L_p$$ Brunn–Minkowski type inequalities

Abstract

Abstract$$L_p$$ L p Brunn–Minkowski type inequalities for the lattice point enumerator $$\mathrm {G}_n(\cdot )$$ G n ( · ) are shown, $$p\ge 1$$ p ≥ 1 , both in a geometrical and in a functional setting. In particular, we prove that $$\begin{aligned}\mathrm {G}_n\bigl ((1-\lambda )\cdot K +_p \lambda \cdot L + (-1,1)^n\bigr )^{p/n}\ge (1-\lambda )\mathrm {G}_n(K)^{p/n}+\lambda \mathrm {G}_n(L)^{p/n} \end{aligned}$$ G n ( ( 1 - λ ) · K + p λ · L + ( - 1 , 1 ) n ) p / n ≥ ( 1 - λ ) G n ( K ) p / n + λ G n ( L ) p / n for any $$K, L\subset \mathbb {R}^n$$ K , L ⊂ R n bounded sets with integer points and all $$\lambda \in (0,1)$$ λ ∈ ( 0 , 1 ) . We also show that these new discrete analogues (for $$\mathrm {G}_n(\cdot )$$ G n ( · ) ) imply the corresponding results concerning the Lebesgue measure.