The Brunn-Minkowski inequality for the generalized Gaussian distribution

Let $\mu_p$ be the generalized Gaussian distribution on $\mathbb{R}^n$ with density $e^{-\frac{|x|^p}{p}}$ 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 $$\mu_p(\lambda K+(1-\lambda) L)^{\alpha_p(n)} \geq \lambda \mu_p(K)^{\alpha_p(n)}+(1-\lambda) \mu_p(L)^{\alpha_p(n)}$$ 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 $n\to +\infty$ is proved.