Topological transversals to a family of convex sets

TL;DR

This paper introduces a topological concept of transversals to convex sets in Euclidean space, linking it to classical transversality and deriving geometric consequences using algebraic topology and combinatorial theorems.

Contribution

It establishes a connection between topological and classical transversals for convex sets and applies advanced topological tools to derive geometric results.

Findings

Topological $ ho$-transversals imply classical $ ho$-transversals under certain conditions.

Uses Schubert cocycles and Lusternik-Schnirelmann category to derive geometric consequences.

Extends colorful Helly theorem to topological transversality context.

Abstract

Let $\mathcal F$ be a family of compact convex sets in $\mathbb R^d$. We say that $\mathcal F $ has a \emph{topological $\rho$-transversal of index $(m,k)$} ($\rho<m$, $0<k\leq d-m$) if there are, homologically, as many transversal $m$-planes to $\mathcal F$ as $m$-planes containing a fixed $\rho$-plane in $\mathbb R^{m+k}$. Clearly, if $\mathcal F$ has a $\rho$-transversal plane, then $\mathcal F$ has a topological $\rho$-transversal of index $(m,k),$ for $\rho<m$ and $k\leq d-m$. The converse is not true in general. We prove that for a family $\mathcal F$ of $\rho+k+1$ compact convex sets in $\mathbb R^d$ a topological $\rho$-transversal of index $(m,k)$ implies an ordinary $\rho$-transversal. We use this result, together with the multiplication formulas for Schubert cocycles, the Lusternik-Schnirelmann category of the Grassmannian, and different versions of the colorful Helly theorem by B\'ar\'any and Lov\'asz, to obtain some geometric consequences.