AI 中文总结
该研究针对凸体的体积比界问题，通过将凸体置于各向同性位置并结合相关估计与新的无维界，改进了已沿用近二十五年的通用体积比上界。
AI 中文摘要
我们证明，对于任意一对凸体 $K,L\subset\mathbb R^n$，体积比 $\operatorname{vr}(K,L)\leq C\sqrt{n\log(n+1)}$。核心在于将 $K$ 和 $L^\circ$ 置于各向同性位置，随后考虑 $L$ 的随机正交像，结合 Bizeul 与 Klartag 的各向同性平均 gauge 估计，以及 Letwin 近期提出的该估计中三阶矩参数的无维界，控制相应算子范数。此结果改进了 Giannopoulos 和 Hartzoulaki 证明的 $\operatorname{vr}(K,L)\leq C\sqrt n \log(n+1)$，该界作为最优通用估计已近二十五年。
英文摘要
We show that, for every pair of convex bodies $K,L\subset\mathbb R^n$, $$ \operatorname{vr}(K,L)\leq C\sqrt{n\log(n+1)}. $$ The main point is to place $K$ and $L^\circ$ in isotropic position. We then consider a random orthogonal image of $L$ and control the corresponding operator norm by combining the isotropic mean-gauge estimate of Bizeul and Klartag with Letwin's recent dimension-free bound for the third-moment parameter appearing in their estimate. Our result improves the bound $ \operatorname{vr}(K,L)\leq C\sqrt n \log(n+1)$ proved by Giannopoulos and Hartzoulaki, which had remained the best general estimate for nearly two and a half decades.
Comments8 pages. References added, together with alternative proofs along the same lines using recent results
