Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On lattice illumination of smooth convex bodies

Lenny Fukshansky

Abstract

The illumination conjecture is a classical open problem in convex and discrete geometry, asserting that every compact convex body~$K$ in $\mathbb R^n$ can be illuminated by a set of no more than $2^n$ points. If $K$ has smooth boundary, it is known that $n+1$ points are necessary and sufficient. We consider an effective variant of the illumination problem for bodies with smooth boundary, where the illuminating set is restricted to points of a lattice and prove the existence of such a set close to $K$ with an explicit bound on the maximal distance. We produce improved bounds on this distance for certain classes of lattices, exhibiting additional symmetry or near-orthogonality properties. Our approach is based on the geometry of numbers.

On lattice illumination of smooth convex bodies

Abstract

The illumination conjecture is a classical open problem in convex and discrete geometry, asserting that every compact convex body~ in can be illuminated by a set of no more than points. If has smooth boundary, it is known that points are necessary and sufficient. We consider an effective variant of the illumination problem for bodies with smooth boundary, where the illuminating set is restricted to points of a lattice and prove the existence of such a set close to with an explicit bound on the maximal distance. We produce improved bounds on this distance for certain classes of lattices, exhibiting additional symmetry or near-orthogonality properties. Our approach is based on the geometry of numbers.
Paper Structure (3 sections, 8 theorems, 47 equations, 2 figures)

This paper contains 3 sections, 8 theorems, 47 equations, 2 figures.

Table of Contents

  1. Introduction and statement of results
  2. Illumination distance
  3. Lattice illumination

Key Result

Proposition 1.1

Let $K \in {\mathcal{K}}_n$, then for any ${\varepsilon} > 0$ there exists $S \subset {\mathbb R}^n$ with $|S| = n+1$ that illuminates $K$ so that

Figures (2)

  • Figure 1: Illustration of the construction in the proof of Lemma \ref{['ball1']}.
  • Figure 2: Illustration of the construction in the proof of Theorem \ref{['main']}.

Theorems & Definitions (8)

  • Proposition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3