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arXiv · 2609.06614

Convex-Hull Instability of the $\gamma_2$-Functional in $L^p$

Helen W. J. Zhang·Chengdong Zhao
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Abstract

For every $2 2$, 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 $2$.

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Helen W. J. Zhang, Chengdong Zhao. 2026-09-06. Convex-Hull Instability of the $\gamma_2$-Functional in $L^p$. https://arxiv.org/abs/2609.06614

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