The Brunn-Minkowski inequality for the generalized Gaussian distribution

Abstract

Let $μ_p$ be the generalized Gaussian distribution on $\mathbb{R}^n$ with density $e^{-\frac{|x|^p}{p}}$ multiplied by a constant depending on $p\ge 1$ and $n$, and $α_p(n)$ be the largest number such that the Brunn-Minkowski type inequality $$μ_p(λK+(1-λ) L)^{α_p(n)} \geq λμ_p(K)^{α_p(n)}+(1-λ) μ_p(L)^{α_p(n)}$$ holds for all convex bodies $K,L$ in $\mathbb{R}^n$ containing the origin and $λ\in[0,1]$. In this paper, the new lower and upper bounds for $α_p(n)$ are found, and their asymptotically optimality as $n\to +\infty$ is proved.