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On discrete log-Brunn-Minkowski type inequalities
(Society for Industrial and Applied Mathematics, 2022) Hernández Cifre, María de los Ángeles; Lucas Marín, Eduardo; Matemáticas
The conjectured log-Brunn-Minkowski inequality says that the volume of centrally symmetric convex bodies K, L ⊂ Rn satisfies vol (1−λ)·K+0 λ·L ≥ vol(K)1−λvol(L)λ, λ ∈ (0, 1), and is known to be true in the plane and for particular classes of n-dimensional symmetric convex bodies. In this paper we get some discrete log-Brunn-Minkowski type inequalities for the lattice point enumerator. Among others, we show that if K, L ⊂ Rn are unconditional convex bodies and λ ∈ (0, 1), then G (1 − λ) · K +C + λ · L+C 1 1 n ≥ G (K)1−λG (L)λ, where Cn = [−1/2, 1/2]n. Neither Cn nor (−1/2, 1/2)n can be removed. Furthermore, it implies the (volume) log-Brunn-Minkowski inequality for unconditional convex bodies. The corresponding results in the Lp setting for 0 < p < 1 are also obtained.