对数凹测度维度Brunn-Minkowski不等式中的普适界

Galyna V. Livshyts

对数凹测度维度Brunn-Minkowski不等式中的普适界

度量几何 2026-05-14 v3 偏微分方程分析概率论

作者: Galyna V. Livshyts

摘要

我们证明了对 $\mathbb{R}^n$ 上任意对数凹测度 $\mu$ 、任意一对对称凸集 $K$ 与 $L$ 以及任意 $\lambda\in [0,1]$ ，有 $\mu((1-\lambda) K+\lambda L)^{c_n}\geq (1-\lambda) \mu(K)^{c_n}+\lambda\mu(L)^{c_n},$ 其中 $c_n\geq n^{-4-o(1)}$ 。这构成了朝向维度Brunn-Minkowski猜想（见Gardner、Zvavitch \cite{GZ}，Colesanti、L、Marsiglietti \cite{CLM}）的进展。此外，我们的界对于各类特殊的对数凹测度有所改进。

关键词

Brunn-Minkowski不等式对数凹测度凸集普适界维度猜想

引用

@article{arxiv.2107.00095,
  title  = {A universal bound in the dimensional Brunn-Minkowski inequality for log-concave measures},
  author = {Galyna V. Livshyts},
  journal= {arXiv preprint arXiv:2107.00095},
  year   = {2026}
}

备注

19 pages; This was initially part of arXiv:2103.11433, but that paper was split into two papers. A minor correction to the exposition was fixed in May 2026. Namely, minor changes were made to Proposition 4.2 and to the argument at the beginning of the proof of Theorem A

对数凹测度维度Brunn-Minkowski不等式中的普适界

摘要

关键词

引用

备注

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