# Covering by homothets and illuminating convex bodies

## TL;DR

This paper investigates coverings and illuminations of convex bodies using homothets, establishing bounds, inequalities, and disproving a conjecture related to the minimal number of illumination directions.

## Contribution

It introduces new bounds for covering and illumination numbers, proves an inequality between them, and disproves a conjecture for the cube case.

## Key findings

- Proved that $g_{\alpha}(B) \leq h_{\alpha}(B)$ for convex bodies.
- Established bounds for $g_{\alpha}(B)$ and $h_{\alpha}(B)$.
- Disproved the conjecture that $h_{\alpha}(B) \leq 2^{d-\alpha}$ for the cube.

## Abstract

The paper is devoted to coverings by translative homothets and illuminations of convex bodies. For a given positive number $\alpha$ and a convex body $B$, $g_{\alpha}(B)$ is the infimum of $\alpha$-powers of finitely many homothety coefficients less than 1 such that there is a covering of $B$ by translative homothets with these coefficients. $h_{\alpha}(B)$ is the minimal number of directions such that the boundary of $B$ can be illuminated by this number of directions except for a subset whose Hausdorff dimension is less than $\alpha$. In this paper, we prove that $g_{\alpha}(B)\leq h_{\alpha}(B)$, find upper and lower bounds for both numbers, and discuss several general conjectures. In particular, we show that $h_{\alpha} (B) > 2^{d-\alpha}$ for almost all $\alpha$ and $d$ when $B$ is the $d$-dimensional cube, thus disproving the conjecture from Research Problems in Discrete Geometry by Brass, Moser, and Pach.