Abstract: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.
| Comments: | 21 pages,5 figures |
| Subjects: | Metric Geometry (math.MG); Probability (math.PR) |
| MSC classes: | 52A40, 60E15 |
| Cite as: | arXiv:2605.24472 [math.MG] |
| (or arXiv:2605.24472v1 [math.MG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24472 arXiv-issued DOI via DataCite (pending registration) |
From: Ge Xiong [view email]
[v1]
Sat, 23 May 2026 08:46:10 UTC (220 KB)
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