A Sharp Surface-Area Extension of Vaaler's Theorem

Abstract

Karasev proved, in dimensions two and three, a sharp surface-area counterpart of a polyhedral extension of Vaaler's theorem [8, Theorem 1.2]. We remove the dimension restriction. More precisely, if an n-dimensional convex polytope P contains the origin in its interior and the affine hull of every nonempty proper face of codimension k∈{1,…,n} is at distance at least √k from the origin, then ℋⁿ⁻¹(∂ P)geqslant n2ⁿ. Consequently, the boundary of every n-dimensional linear section of the cube [−1,1]^N has (n−1)-dimensional measure at least n2ⁿ. This confirms, in all dimensions, a conjecture of Grigory M. Ivanov recorded by Karasev [8, Section 1]. The proof combines the Rogers--Karasev flag decomposition with a dimension-free Gaussian comparison for orthoschemes. Its main step is an ordered-square substitution which converts all face-distance assumptions into a pointwise domination of a single positive integral. We also disprove the naive extension to all skeletal measures, formulate a ridge-skeleton conjecture, and establish two partial results toward it.

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BibTeX

@misc{zwierzynski2026sharp,
  title = {{A Sharp Surface-Area Extension of Vaaler's Theorem}},
  author = {Michał Zwierzyński},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.27798},
  eprint = {2609.27798},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27798},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - A Sharp Surface-Area Extension of Vaaler's Theorem
AU  - Michał Zwierzyński
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.27798
UR  - https://arxiv.org/abs/2609.27798
AB  - Karasev proved, in dimensions two and three, a sharp surface-area counterpart of a polyhedral extension of Vaaler's theorem [8, Theorem 1.2]. We remove the dimension restriction. More precisely, if an n-dimensional convex polytope P contains the origin in its interior and the affine hull of every nonempty proper face of codimension k∈{1,…,n} is at distance at least √k from the origin, then ℋ^n−1(∂ P)geqslant n2^n. Consequently, the boundary of every n-dimensional linear section of the cube [−1,1]^N has (n−1)-dimensional measure at least n2^n. This confirms, in all dimensions, a conjecture of Grigory M. Ivanov recorded by Karasev [8, Section 1]. The proof combines the Rogers--Karasev flag decomposition with a dimension-free Gaussian comparison for orthoschemes. Its main step is an ordered-square substitution which converts all face-distance assumptions into a pointwise domination of a single positive integral. We also disprove the naive extension to all skeletal measures, formulate a ridge-skeleton conjecture, and establish two partial results toward it.
ER  -

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