The Brunn-Minkowski inequality for the generalized Gaussian distribution
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918519868751872 |
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| author | Xiong, Ge Yang, Kai-Wen |
| author_facet | Xiong, Ge Yang, Kai-Wen |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_24472 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Brunn-Minkowski inequality for the generalized Gaussian distribution Xiong, Ge Yang, Kai-Wen Metric Geometry Probability 52A40, 60E15 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. |
| title | The Brunn-Minkowski inequality for the generalized Gaussian distribution |
| topic | Metric Geometry Probability 52A40, 60E15 |
| url | https://arxiv.org/abs/2605.24472 |