A New Topological Helly Theorem and some Transversals Results

TL;DR

This paper introduces a new topological Helly theorem based on homology conditions and applies it to derive novel results on transversals of convex sets, expanding classical geometric intersection theory.

Contribution

The paper presents a new topological Helly theorem involving homology conditions and uses it to obtain new results on affine transversals to convex sets.

Findings

Established a topological Helly theorem with homology conditions.

Derived new results on affine transversals to convex sets.

Extended classical intersection theorems using topological methods.

Abstract

We prove that for a topological space X with the property that $H_p(U)=0$ for $p\geq d$ and every open subset $U$ of $X$, a finite family of open sets in $X$ has nonempty intersection if for any subfamily of size $j$, $1\leq j \leq d+1$, the $(d-j)$-dimensional homology group of its intersection is zero. We use this theorem to prove new results concerning transversal affine planes to families of convex sets.