Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Alexey Garber

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 $k+6$.

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

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 copies of a single lattice the Helly number does not exceed .

Paper Structure

This paper contains 4 sections, 13 theorems, 9 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Helly numbers for arbitrary sets
  3. Crystals and cut-and-project sets
  4. Helly numbers for two-dimensional crystals

Key Result

Theorem 1.1

Let $\mathcal{F}$ be a finite family of convex sets in $\mathbb{R}^d$. If every $2^d$ sets from $\mathcal{F}$ intersect at a point of $\mathbb{Z}^d$, then all sets from $\mathcal{F}$ intersect at a point of $\mathbb{Z}^d$.

Figures (5)

  • Figure 1: Cut-and-project construction of the Fibonacci tiling. The bold vertical segment represents the window, the shaded strip "cuts" the points with $\pi_2$-projections in the window, and the solid points on the horizontal axis are the $\pi_1$-projections of the points in the strip.
  • Figure 2: Penrose and Ammann-Beenker tilings. The images are courtesy of Dirk Frettlöh.
  • Figure 3: Two-dimensional $6$-crystal with Helly number $12$. The red arcs allow construction of $k$-crystals with Helly number $k+6$ for every $k\geq 6$.
  • Figure 4: $P$ has three vertices from two copies of $\mathbb{Z}^2$.
  • Figure 5: $4$-crystal with $h(S)=9$.

Theorems & Definitions (24)

  • Theorem 1.1: J.-P. Doignon, Doi
  • Theorem 1.2: I. Bárány, J. Matoušek, BM
  • Definition 2.1
  • Lemma 2.2: G. Averkov, Ave
  • Definition 3.1
  • Lemma 3.2
  • Corollary 3.3
  • proof
  • Definition 3.4
  • Theorem 3.5
  • ...and 14 more