PreprintCC BY 4.0arXiv
KKM meets colorful Carathéodory
Abstract
The colorful Carathéodory theorem states that if V1,…,Vd+1⊂ℝd satisfy 0∈conv Vi for every i, then there exist points vi∈ Vi such that 0∈conv {v1,…,vd+1}. We give a new proof based on triangulations and the classical KKM lemma. We lift the sets Vi to parallel affine copies of ℝd in ℝd×ℝd and use a triangulation of the resulting point configuration to construct a KKM cover. Using the same approach, we prove the very colorful Carathéodory theorem.
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BibTeX
@misc{polyanskii2026meets,
title = {{KKM meets colorful Carathéodory}},
author = {Alexander Polyanskii},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22829},
eprint = {2610.22829},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22829},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - KKM meets colorful Carathéodory
AU - Alexander Polyanskii
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.22829
UR - https://arxiv.org/abs/2610.22829
AB - The colorful Carathéodory theorem states that if V_1,…,V_d+1⊂ℝ^d satisfy 0∈conv V_i for every i, then there exist points v_i∈ V_i such that 0∈conv {v_1,…,v_d+1}. We give a new proof based on triangulations and the classical KKM lemma. We lift the sets V_i to parallel affine copies of ℝ^d in ℝ^d×ℝ^d and use a triangulation of the resulting point configuration to construct a KKM cover. Using the same approach, we prove the very colorful Carathéodory theorem.
ER -