bibkey: barany1982colourful
authors: Imre Bárány
year: 1982
title: A generalization of Carathéodory's theorem
doi: 10.1016/0012-365X(82)90115-7
url: https://users.renyi.hu/~barany/cikkek/9.pdf
claim: Theorem 2.1 gives one representative from each of d+1 arbitrary subsets of real d-space whose convex hulls have a common point, with that point in the selected convex hull.
strata_touched:
  - D5/S3/Geometry/ColourfulCaratheodory
license: citation-only
triage: anchor

Colourful Carathéodory selection

Imre Bárány, A generalization of Carathéodory’s theorem, Discrete Mathematics 40 (1982), 141–152, Theorem 2.1 on printed page 141. The sets in that theorem are arbitrary subsets of real Euclidean space. Their convex hulls contain a common point, and the conclusion selects one vector from every labelled set while retaining that point in the selected convex hull. The statement imposes no finiteness, closedness, boundedness, general position, disjointness or distinctness assumption.

The nearest-point exchange proof uses finite subsets internally, obtained from each original convex hull membership by ordinary Carathéodory. A nearest nonzero point in a colourful hull has a positive affine support on its supporting hyperplane. That support is linearly independent and uses at most d labels. A point of an omitted colour lies strictly across the supporting level, so replacing that colour gives a strictly closer point in the new hull. A minimum among the finitely many finite-core colourful hulls therefore equals the common point. Translation treats an arbitrary common point. The formal statement also includes dimension zero. These are classical convex-geometric conclusions; no originality is claimed.

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Verified locator

DOI: 10.1016/0012-365X(82)90115-7

https://doi.org/10.1016/0012-365X(82)90115-7

https://users.renyi.hu/~barany/cikkek/9.pdf

Theorem 2.1, printed page 141 (PDF page 1).