The Brunn-Minkowski inequality for the generalized Gaussian distribution

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