AI 中文总结
研究高斯多胞体与标准交叉多胞体的Banach-Mazur距离，证明当m=n^3时距离至少为c n^{5/8}(ln n)^{-5/8}的概率趋于1，改进了Friedland的指数4/7下界，方法结合离散化、条件化及K/U分解。
AI 中文摘要
设 $B_1^n$ 为 $\mathbb R^n$ 中的标准交叉多胞体，设 $g_1,\ldots,g_m$ 为 $\mathbb R^n$ 中独立的标准化高斯向量，并令 $G_m=\operatorname{conv}\{\pm g_1,\ldots,\pm g_m\}$。对于 $m=n^3$，证明了对适当的绝对常数 $c>0$，有 $$ \mathbb P\left\{d_{\mathrm{BM}}(G_m,B_1^n)\geqslant c n^{5/8}(\ln n)^{-5/8}\right\}\geqslant1-\frac2n $$。这改进了Friedland最近下界中的指数 $4/7$。证明使用了Friedland的离散化和条件化论证以及 $K/U$ 分解。抑制选定的 $K$ 向量族，并对剩余的 $K$ 向量取商。在所得商空间中，对由被抑制的 $K$ 向量和 $U$ 向量形成的每个最高维外积证明了同时界。在Löwner归一化后的Dvoretzky-Rogers选择将这些行列式估计转化为整个投影多胞体的最小体积椭球体的界，而Maurey的经验方法随后给出所需的高斯测度估计。
英文摘要
Let $B_1^n$ be the standard cross-polytope in $\mathbb R^n$, let $g_1,\ldots,g_m$ be independent standard Gaussian vectors in $\mathbb R^n$, and set $G_m=\operatorname{conv}{\pm g_1,\ldots,\pm g_m}$. For $m=n^3$ it is proved that $$ \mathbb P\left\{d_{\mathrm{BM}}(G_m,B_1^n)\geqslant c n^{5/8}(\ln n)^{-5/8}\right\}\geqslant 1-\frac2n $$ for a suitable absolute constant $c>0$. This independently improves the polynomial exponent $4/7$ in Friedland's preceding work. Independent concurrent work of Friedland, which appeared after completion of the present manuscript, obtains the same polynomial exponent with the stronger logarithmic factor $(\ln n)^{-1/4}$ by a different argument. The proof uses Friedland's discretization and conditioning argument together with the $K/U$ decomposition. A selected family of $K$ vectors is suppressed and the remaining $K$ vectors are quotiented out. In the resulting quotient simultaneous bounds are proved for every top-dimensional exterior product formed from the suppressed $K$ vectors and the $U$ vectors. A Dvoretzky-Rogers selection after L"owner normalization converts these determinant estimates into a bound for the minimum volume ellipsoid of the whole projected polytope and Maurey's empirical method then gives the required Gaussian measure estimate.
Commentsv2: Added discussion of independent concurrent work of Friedland (arXiv:2608.17743), including the chronology and a comparison of the two proofs. The main theorem and proof are unchanged
