Tverberg cores and Kalai's cascade conjecture

TL;DR

This paper explores topological versions of Kalai's cascade conjecture, establishing conditions under which certain Tverberg points persist after vertex deletions and confirming the conjecture for specific finite point sets.

Contribution

It proves a topological analogue of a consequence of Kalai's cascade conjecture for prime power r and confirms the conjecture for Radon sets with 0-dimensional Radon sets.

Findings

Proved existence of points remaining Tverberg points after vertex deletions under specified conditions.

Established the topological analogue of a cascade conjecture for prime power r.

Confirmed the cascade conjecture for finite point sets with 0-dimensional Radon sets.

Abstract

We study topological analogues of Kalai's cascade conjecture. Given a continuous map from an $n$-simplex to $\mathbb R^d$, let $T_r(f)$ be the set of points contained in the images of $r$ pairwise disjoint faces. We prove that if $r$ is a prime power and $\dim T_r(f)\le k$, then there exists a point that remains an $r$-Tverberg point after any $t$ vertices are deleted, provided $n=(r-1)(d+1)+t(k+1)$. For $t=1$, this gives a topological analogue of a standard consequence of Kalai's cascade conjecture. We also confirm the cascade conjecture for finite point sets whose Radon set is $0$-dimensional.