On affine Tverberg-type results without continuous generalization

TL;DR

This paper demonstrates fundamental differences between affine and continuous Tverberg-type results, providing elementary proofs, counterexamples, and settling the AP conjecture negatively, highlighting the distinct nature of affine and topological intersection patterns.

Contribution

It offers elementary proofs of affine-continuous differences, disproves the AP conjecture, and extends Tverberg-type results without divisibility conditions, emphasizing the non-equivalence of affine and continuous cases.

Findings

Affine and continuous Tverberg results differ fundamentally.

The AP conjecture is false in general.

Affine Tverberg results can hold without divisibility conditions.

Abstract

Recent progress building on the groundbreaking work of Mabillard and Wagner has shown that there are important differences between the affine and continuous theory for Tverberg-type results. These results aim to describe the intersection pattern of convex hulls of point sets in Euclidean space and continuous relaxations thereof. Here we give additional examples of an affine-continuous divide, but our deductions are almost elementary and do not build on the technical work of Mabillard and Wagner. Moreover, these examples show a difference between the affine and continuous theory even asymptotically for arbitrarily large complexes. Along the way we settle the Tverberg admissible-prescribable problem (or AP conjecture) in the negative, give a new, short and elementary proof of the balanced case of the AP conjecture which was recently proven by Joji\'c, Vre\'cica, and \v{Z}ivaljevi\'c in a series of two papers, provide examples of Tverberg-type results that hold affinely but not continuously without divisibility conditions on the intersection multiplicity, extend a result of Sober\'on, and show that this extension has a topological generalization if and only if the intersection multiplicity is a prime.