Dvoretzky定理中的一个多项式界
A polynomial bound in Dvoretzky's theorem
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中文总结 AI 辅助
本文给出ε-Dvoretzky猜想的简单概率证明，表明Dvoretzky定理中对逼近参数ε的依赖为多项式，并得到更尖锐的界及有限凸体族的同步版本，同时将逼近球半径与平均宽度等几何参数比较。
中文摘要 AI 辅助
我们给出了ε-Dvoretzky猜想的一个简单证明，该猜想断言在Dvoretzky定理中，对逼近参数ε的依赖是关于1/ε的多项式。特别地，若n≥(C/ε)^{ℓ/2+1}，则任意n维凸体通过任意给定内点都有一个ℓ维截面，该截面与欧氏球ε-接近。这里C>0是一个普适常数。实际上，我们获得了对ε更尖锐的依赖关系。该证明是概率性的，但使用了与先前不同的概率模型。我们还证明了我们的定理对有限个以原点为内点的凸体族的同步版本，得到一个公共线性子空间，使得所有凸体在该子空间上都有近似欧氏的截面。最后，我们将逼近欧氏球的半径与熟悉的几何参数（如凸体及其对偶的平均宽度）进行比较。
英文摘要
We present a simple proof of the $\varepsilon$-Dvoretzky conjecture, which asserts that the dependence on the approximation parameter $\varepsilon$ in Dvoretzky's theorem is polynomial in $1/\varepsilon$. In particular, if $n \geq (C/\varepsilon)^{\ell/2+1}$, then any $n$-dimensional convex body has, through any given interior point, an $\ell$-dimensional section that is $\varepsilon$-close to a Euclidean ball. Here, $C > 0$ is a universal constant. We in fact obtain a sharper dependence on $\varepsilon$. The proof is probabilistic, but uses a different probabilistic model from those employed previously. We also prove a simultaneous version of our theorem for a finite family of convex bodies with the origin in their interior, yielding a common linear subspace on which all of the bodies have nearly-Euclidean sections. Finally, we compare the radius of the approximating Euclidean ball with familiar geometric parameters, such as the mean widths of the body and its dual.
