凸体的最优平均宽度与度量熵估计
Optimal mean width and metric entropy estimates for convex bodies
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中文总结 AI 辅助
该研究针对凸体，证明了球面平均宽度与体积半径比值的最优界，以及欧氏覆盖数对数的最大值由单纯形和交叉多面体取得，证明采用了 Eldan 的随机局部化方法。
中文摘要 AI 辅助
我们证明存在常数 $C>0$，使得对任意 $n\geq1$ 和任意凸体 $K\subset\mathbf{R}^n$，有 $1\leq\inf_{T\in\mathrm{GL}(n)}\frac{M^\ast(TK)}{\mathrm{vr}(TK)}\leq C\sqrt{\log(\mathrm{e}n)}$，其中 $M^\ast$ 表示球面平均宽度，$\mathrm{vr}(\cdot)$ 表示体积半径。右侧的上界在相差通用常数的意义下，由交叉多面体和正则 $n$ 单纯形达到。类似地，我们证明在相差通用常数的意义下，欧氏覆盖数的对数在 $\mathbf{R}^n$ 中的凸体上由单纯形和交叉多面体取到最大值。我们的证明用到了 Eldan 的随机局部化方法。
英文摘要
We show that for any $n \geq 1$ and any convex body $K \subset \mathbf{R}^n$, there exists $T \in \mathrm{SL}(n)$ such that \[ \mathrm{diam}(TK) \lesssim \sqrt{n} \mathrm{vr}(K), \quad \mbox{and} \quad M^\ast\Big(T(K-x) \cap r\mathrm{vr}(K)\,B^n_2\Big)\lesssim \mathrm{vr}(K) \sqrt{\log(\mathrm{e} r^2)}, \] for every $x \in K$ and every $r \geq 1$. Above, $M^\ast(\cdot)$ denotes the spherical mean width and $\mathrm{vr}(\cdot)$ denotes the volume radius. As a consequence, we establish for any convex body $K \subset \mathbf{R}^n$ that \[ 1 \leq \inf_{T \in \mathrm{SL}(n)} \, \frac{M^\ast(TK)}{\mathrm{vr}(K)} \lesssim \sqrt{\log(\mathrm{e} n)}. \] The estimates above are sharp, up to universal constants, as they are attained for the crosspolytope and any regular $n$-simplex. Up to universal constants, our results show that all quermassintegrals, and the logarithm of the covering numbers for all scales, simultaneously for $K$ and the polar body $K^\circ$, are maximized by the simplex and crosspolytope. Our proof makes use of Eldan's stochastic localization. To establish the results, we work with an extension of Bobkov's maximal Gaussian measure position to possibly non-symmetric convex bodies. Our results imply that this position is an optimal "regular" Milman position, thereby improving a result of G. Pisier. We establish a non-symmetric analogue of the strong Gaussian (B)-theorem which may be of independent interest.
发表机构
- Texas A&M University（德克萨斯农工大学）
- Princeton University（普林斯顿大学）
- UC Berkeley（加州大学伯克利分校）
- Cornell University（康奈尔大学）
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