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arXiv · 2609.21927

A colorful Steinitz theorem with different centers

G. Ivanov
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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 $\R^d$ 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 $1 \le k \le d-1$, if every rainbow selection from $2d-k+1$ finite families of convex sets in $\mathbb{R}^d$ has a $k$-dimensional cone in its intersection, then the intersection of one of the families contains such a cone.

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G. Ivanov. 2026-09-18. A colorful Steinitz theorem with different centers. https://arxiv.org/abs/2609.21927

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