On Helly number for crystals and cut-and-project sets

TL;DR

This paper establishes the existence of Helly numbers for crystals and cut-and-project sets with convex windows, providing bounds for two-dimensional crystals composed of multiple lattice copies.

Contribution

It introduces the concept of Helly numbers for these geometric structures and derives an upper bound for two-dimensional crystals with multiple lattice copies.

Findings

Helly numbers exist for crystals and cut-and-project sets with convex windows.

For 2D crystals with k lattice copies, the Helly number is at most k+6.

The paper provides new bounds and existence results for Helly numbers in these settings.

Abstract

We prove existence of Helly numbers for crystals and for cut-and-project sets with convex windows. Also we show that for a two-dimensional crystal consisting of $k$ copies of a single lattice the Helly number does not exceed $k+6$.