A universal bound in the dimensional Brunn-Minkowski inequality for log-concave measures
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866913120411189248 |
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| author | Livshyts, Galyna V. |
| author_facet | Livshyts, Galyna V. |
| contents | We show that for any log-concave measure $μ$ on $\mathbb{R}^n$, any pair of symmetric convex sets $K$ and $L$, and any $λ\in [0,1],$ $$μ((1-λ) K+λL)^{c_n}\geq (1-λ) μ(K)^{c_n}+λμ(L)^{c_n},$$ where $c_n\geq n^{-4-o(1)}.$ This constitutes progress towards the dimensional Brunn-Minkowski conjecture (see Gardner, Zvavitch \cite{GZ}, Colesanti, L, Marsiglietti \cite{CLM}). Moreover, our bound improves for various special classes of log-concave measures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_00095 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A universal bound in the dimensional Brunn-Minkowski inequality for log-concave measures Livshyts, Galyna V. Metric Geometry Analysis of PDEs Probability We show that for any log-concave measure $μ$ on $\mathbb{R}^n$, any pair of symmetric convex sets $K$ and $L$, and any $λ\in [0,1],$ $$μ((1-λ) K+λL)^{c_n}\geq (1-λ) μ(K)^{c_n}+λμ(L)^{c_n},$$ where $c_n\geq n^{-4-o(1)}.$ This constitutes progress towards the dimensional Brunn-Minkowski conjecture (see Gardner, Zvavitch \cite{GZ}, Colesanti, L, Marsiglietti \cite{CLM}). Moreover, our bound improves for various special classes of log-concave measures. |
| title | A universal bound in the dimensional Brunn-Minkowski inequality for log-concave measures |
| topic | Metric Geometry Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2107.00095 |