PreprintCC BY 4.0arXiv
Variance lower bounds for geometric functionals of rotationally invariant log-concave random polytopes
Abstract
We establish variance lower bounds for the intrinsic volumes and the face numbers of random polytopes, generated by independent samples from rotationally invariant log-concave probability measures on ℝd with full support. Our results extend the corresponding Gaussian lower bounds of Bárány and Vu and of Bárány and Thäle to this broader class of distributions. Our proof builds on the geometric construction introduced by Bárány and Vu, for which we develop an alternative treatment based on barycentric coordinates. We combine it with local variance estimates for both intrinsic volumes and the number of faces.
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BibTeX
@misc{pafis2026variance,
title = {{Variance lower bounds for geometric functionals of rotationally invariant log-concave random polytopes}},
author = {Minas Pafis and Christos Pandis},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.21663},
eprint = {2608.21663},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.21663},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Variance lower bounds for geometric functionals of rotationally invariant log-concave random polytopes
AU - Minas Pafis
AU - Christos Pandis
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.21663
UR - https://arxiv.org/abs/2608.21663
AB - We establish variance lower bounds for the intrinsic volumes and the face numbers of random polytopes, generated by independent samples from rotationally invariant log-concave probability measures on ℝ^d with full support. Our results extend the corresponding Gaussian lower bounds of Bárány and Vu and of Bárány and Thäle to this broader class of distributions. Our proof builds on the geometric construction introduced by Bárány and Vu, for which we develop an alternative treatment based on barycentric coordinates. We combine it with local variance estimates for both intrinsic volumes and the number of faces.
ER -