Topological transversals to a family of convex sets

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

On a transversal theorem of Montejano and Karasev

“…While the result in [5] uses technical algebraic and topological tools, our proof is a simple application of the Borsuk-Ulam theorem. Additionally, in certain cases we obtain stronger results than those in [5].…”

“…The paper by Montejano and Karasev [5] contains several other interesting results that are not covered by our Theorem 1. In particular, Corollary 7 in [5] gives conditions for when families of size greater than k + 2 have a k-transversal.…”

“…The paper by Montejano and Karasev [5] contains several other interesting results that are not covered by our Theorem 1. In particular, Corollary 7 in [5] gives conditions for when families of size greater than k + 2 have a k-transversal. For example they show the following: We have not been able to prove the statement above using the proof method of Theorem 1.…”

“…These observations have led to a number of interesting results and conjectures concerning families satisfying the colorful intersection property [3,4,5]. Here we focus on the remarkable results by Montejano and Karasev [5], which state that under certain conditions on n, |F i | and d, the colorful intersection property implies that that one of the families F i admits a k-transversal, that is, a k-dimensional affine flat that intersects every memebr of F i . In its simplest non-trivial form, their theorem can be formulated as follows (See also [4,Theorem 3.2]): Consider 3 red convex sets and 3 blue convex sets in R 3 , and suppose every red set intersects every blue set.…”

A parametrized version of the Borsuk–Ulam–Bourgin–Yang–Volovikov theorem

“…These parametrized versions are then used to prove cases of the so-called transversal versions of Tverberg's theorem, the Tverberg-Vrećica conjecture [TV93]. More transversal versions of standard theorems in discrete geometry can be found in Karasev [Kar07], [Kar09b], and Montejano-Karasev [MK11] 3 Topological notations Let us fix some notation that will be used throughout the paper.…”