发表机构
School of Mathematics, Hunan University; School of Mathematics and Statistics, Central South University（湖南大学数学学院; 中南大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造有限集证明 $L^p$ 中 $\gamma_2$ 泛函的凸包不稳定性，推翻 Talagrand 问题中的两个估计，并给出尖锐凸化轮廓。
AI 中文摘要
对于每个 $2<p<\infty$ 和每个整数 $r\ge2$，我们构造一个有限集 $T\subseteq S_{L^p[0,1]}$，使得 $|T|\le2^{Cr^2}$，$\gamma_2(T)\le Cr$，且 $\gamma_2(\conv T)\ge c r^{3/2-1/p}$。因此，对于每个固定的 $p>2$，Talagrand 研究问题 2.11.3 中的两个估计在空间 $L^p[0,1]$ 和 $\ell_p$ 中均失效。下界来自应用于欧几里得单纯形的尺度分离乘积的多级乘积原理。同样的构造给出了 $\ell_p^r(\ell_2^{2^{128r}})$ 在 $2<p\le\infty$ 时的尖锐凸化轮廓，并且通过有限表示性，在每一个余型指标大于 2 的无限维巴拿赫空间中定量失效。
英文摘要
For every $2<p<\infty$ and every integer $r\ge2$, we construct a finite set $T\subseteq S_{L^p[0,1]}$ such that $|T|\le2^{Cr^2}$, $γ_2(T)\le Cr$, and $γ_2(\conv T)\ge c r^{3/2-1/p}$. Consequently, for every fixed $p>2$, both estimates in Talagrand's Research Problem~2.11.3 fail in each of the spaces $L^p[0,1]$ and $\ell_p$. The lower bound follows from a multilevel product principle applied to a scale-separated product of Euclidean simplices. The same construction gives the sharp convexification profile of $\ell_p^r(\ell_2^{2^{128r}})$ for $2<p\le\infty$ and, by finite representability, quantitative failure in every infinite-dimensional Banach space with cotype index greater than $2$.
CommentsCurrent frontier generative-AI models can produce essentially similar constructions when presented with the underlying problem. The authors do not plan to submit this work to a journal
