Helly theorem for affine spaces without dimensions

TL;DR

This paper extends the no-dimensional Helly theorem to affine spaces and convex sets, using an unboundedness framework, and proves the optimality of this generalization.

Contribution

It generalizes the no-dimensional Helly theorem to affine spaces and convex sets, establishing its optimality and broadening its applicability.

Findings

Generalization of Helly theorem to affine spaces

Proof of the theorem's optimality

Extension of previous results to convex sets

Abstract

We prove a no-dimensional Helly theorem for affine spaces and convex sets using the unboundedness framework of Aronov, Goodman, and Pollack (Computational Geometry, 2002). This generalizes the fundamental result of Adiprasito, B\'ar\'any, Mustafa, and Terpai on the no-dimensional Helly theorem for points and convex sets (Discrete & Computational Geometry, 2020). Additionally, we establish the optimality of our result.