Family of closed convex sets covering faces of a simplex

TL;DR

This paper explores a convex variant of Sperner's lemma, focusing on covering faces of a simplex with closed convex sets, offering new insights into combinatorial topology.

Contribution

It introduces a novel convex approach to Sperner's lemma, extending classical combinatorial results to convex geometric settings.

Findings

Established conditions for covering faces of a simplex with convex sets

Proved new convex Sperner-type lemmas

Extended combinatorial topology results to convex geometry

Abstract

It is considered a special, convex variant of Sperner lemma type .