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A point in the interior of the convex hulls

Imre Bárány, Yun Qi

TL;DR

The paper proves a colourful extension of Steinitz's theorem: given a system of $2d$ sets $X_1,\ldots,X_{2d}$ in $\mathbb{R}^d$ with each $\text{pos}\,X_i=\mathbb{R}^d$, there exists a transversal $T$ with $\text{pos}\,T=\mathbb{R}^d$. It develops a two-pronged proof strategy: leveraging cone Carathéodory-type lemmas and transversal constructions to obtain a bound of $2d$ elements, and then classifying when equality is forced by identifying two critical configurations, the Basis Case (BCase) and Positive Basis Case (PCase). A key component is an inductive framework on dimension, aided by projections and a detailed case analysis (Cases I and II) that ultimately shows equality occurs precisely in BCase or PCase. The results clarify when the optimal $2d$-set bound is necessary in the colourful setting and extend Steinitz-type results to a transversal/colourful context with precise equality characterizations.

Abstract

Steinitz's theorem states that if a point $a \in \mathrm{int\,conv\,} X$ for a set $X \subset \mathbb{R}^d$, then $X$ contains a subset $Y$ of size at most $2d$ such that $a \in \mathrm{int\,conv\,}Y$. The bound $2d$ is best possible here. We prove the colourful version of this theorem and characterize the cases when exactly $2d$ sets are needed.

A point in the interior of the convex hulls

TL;DR

The paper proves a colourful extension of Steinitz's theorem: given a system of sets in with each , there exists a transversal with . It develops a two-pronged proof strategy: leveraging cone Carathéodory-type lemmas and transversal constructions to obtain a bound of elements, and then classifying when equality is forced by identifying two critical configurations, the Basis Case (BCase) and Positive Basis Case (PCase). A key component is an inductive framework on dimension, aided by projections and a detailed case analysis (Cases I and II) that ultimately shows equality occurs precisely in BCase or PCase. The results clarify when the optimal -set bound is necessary in the colourful setting and extend Steinitz-type results to a transversal/colourful context with precise equality characterizations.

Abstract

Steinitz's theorem states that if a point for a set , then contains a subset of size at most such that . The bound is best possible here. We prove the colourful version of this theorem and characterize the cases when exactly sets are needed.
Paper Structure (6 sections, 8 theorems, 2 equations, 3 figures)

This paper contains 6 sections, 8 theorems, 2 equations, 3 figures.

Table of Contents

  1. Introduction and some background
  2. Proof of Theorem \ref{['th:stein']}
  3. Proof of Theorem \ref{['th:colstein']}
  4. Preparations for the proof of Theorem \ref{['th:main']}
  5. Proof of Theorem \ref{['th:main']} in Case I
  6. Proof of Theorem \ref{['th:main']} in Case II

Key Result

Theorem 1.1

If $X \subset S^{d-1}$ and $\mathrm{pos\,} X=\mathbb{R}^d$, then there is $Y \subset X$ with $\mathrm{pos\,} Y=\mathbb{R}^d$ and $|Y|\le 2d$. Further there is such a $Y$ with $|Y|\le 2d-1$ unless $X=\{\pm e_1,\ldots,\pm e_d\},$ where $e_1,\ldots,e_d$ is a basis of $\mathbb{R}^d$.

Figures (3)

  • Figure 1: The Positive Basis case for $d=2$.
  • Figure 2: $v \in \mathrm{int\,} C$ and $-v \in \mathrm{int\,} C^*$.
  • Figure 3: The small red ball $B$ is contained in $\mathrm{int\,conv\,} C$.

Theorems & Definitions (8)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 3.1
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 6.1