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PreprintCC0arXiv

Convex-Hull Instability of the γ2-Functional in Lp

  • Helen W. J. Zhang
  • Chengdong Zhao
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Abstract

For every 2<p<∞ and every integer r≥2, we construct a finite set T⊆ SLp[0,1] such that |T|≤2Cr2, γ2(T)≤ Cr, and γ2(conv T)≥ c r3/2−1/p. Consequently, for every fixed p>2, both estimates in Talagrand's Research Problem~2.11.3 fail in each of the spaces Lp[0,1] and ℓp. 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 ℓpr(ℓ22128r) for 2<p≤∞ and, by finite representability, quantitative failure in every infinite-dimensional Banach space with cotype index greater than $2$.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026convexhull,
  title = {{Convex-Hull Instability of the $γ_2$-Functional in $L^p$}},
  author = {Helen W. J. Zhang and Chengdong Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06614},
  eprint = {2609.06614},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06614},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - Convex-Hull Instability of the γ_2-Functional in L^p
AU  - Helen W. J. Zhang
AU  - Chengdong Zhao
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.06614
UR  - https://arxiv.org/abs/2609.06614
AB  - For every 2<p<∞ and every integer r≥2, we construct a finite set T⊆ S_L^p[0,1] such that |T|≤2^Cr^2, γ_2(T)≤ Cr, and γ_2(conv T)≥ 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 ℓ_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 ℓ_p^r(ℓ_2^2^128r) for 2<p≤∞ and, by finite representability, quantitative failure in every infinite-dimensional Banach space with cotype index greater than $2$.
ER  -