PreprintarXiv licencearXiv
A colorful quantitative Helly theorem for volume
Abstract
We prove a colorful quantitative Helly theorem for volume with the optimal number 2d of colors. If every rainbow intersection from 2d finite families of convex sets in Rd has volume at least one, then the intersection of one family has volume at least d−O(d2). We also prove a colorful quantitative Steinitz theorem for origin-centered ellipsoids of different shapes. The proof uses a common normalization of positive operators and a lift that produces two rainbow bases with large determinants.
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BibTeX
@misc{ivanov2026colorful,
title = {{A colorful quantitative Helly theorem for volume}},
author = {Grigory Ivanov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25671},
eprint = {2609.25671},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25671},
}
RIS
TY - UNPB
TI - A colorful quantitative Helly theorem for volume
AU - Grigory Ivanov
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.25671
UR - https://arxiv.org/abs/2609.25671
AB - We prove a colorful quantitative Helly theorem for volume with the optimal number 2d of colors. If every rainbow intersection from 2d finite families of convex sets in R^d has volume at least one, then the intersection of one family has volume at least d^−O(d^2). We also prove a colorful quantitative Steinitz theorem for origin-centered ellipsoids of different shapes. The proof uses a common normalization of positive operators and a lift that produces two rainbow bases with large determinants.
ER -