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A colorful Steinitz theorem with different centers
Abstract
We prove a colorful quantitative Steinitz theorem in which the color classes may have different centers. If the convex hull of each of 2d sets in Rd contains a translate of the Euclidean unit ball, then a rainbow convex hull contains a ball of radius (3d)−2d2 whose center belongs to the convex hull of the given centers. The main ingredient is an exact result for translates of a segment, proved by a lifting and a topological colorful Helly theorem. The same idea also yields a sharp colorful Helly theorem for translated cones: for 1 ≤ k ≤ d−1, if every rainbow selection from 2d−k+1 finite families of convex sets in ℝd has a k-dimensional cone in its intersection, then the intersection of one of the families contains such a cone.
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BibTeX
@misc{ivanov2026colorful,
title = {{A colorful Steinitz theorem with different centers}},
author = {G. Ivanov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21927},
eprint = {2609.21927},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21927},
}
RIS
TY - UNPB
TI - A colorful Steinitz theorem with different centers
AU - G. Ivanov
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21927
UR - https://arxiv.org/abs/2609.21927
AB - We prove a colorful quantitative Steinitz theorem in which the color classes may have different centers. If the convex hull of each of 2d sets in R^d contains a translate of the Euclidean unit ball, then a rainbow convex hull contains a ball of radius (3d)^−2d^2 whose center belongs to the convex hull of the given centers. The main ingredient is an exact result for translates of a segment, proved by a lifting and a topological colorful Helly theorem. The same idea also yields a sharp colorful Helly theorem for translated cones: for 1 ≤ k ≤ d−1, if every rainbow selection from 2d−k+1 finite families of convex sets in ℝ^d has a k-dimensional cone in its intersection, then the intersection of one of the families contains such a cone.
ER -