Covering by homothets and illuminating convex bodies

Authors

Alexey Glazyrin, The University of Texas Rio Grande Valley

Document Type

Article

Publication Date

11-15-2021

Abstract

The paper is devoted to coverings by translative homothets and illuminations of convex bodies. For a given positive number α and a convex body B, gα⁡(B) is the infimum of α-powers of finitely many homothety coefficients less than 1 such that there is a covering of B by translative homothets with these coefficients. hα⁡(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 α. In this paper, we prove that gα⁡(B)≤hα⁡(B), find upper and lower bounds for both numbers, and discuss several general conjectures. In particular, we show that hα⁡(B)>2d−α for almost all α and d when B is the d-dimensional cube, thus disproving the conjecture from Brass, Moser, and Pach [Research problems in discrete geometry, Springer, New York, 2005].

Comments

First published in Proc. Amer. Math. Soc. 150 (2022), 779-793, published by the American Mathematical Society. © 2021 American Mathematical Society.

Recommended Citation

Glazyrin, A. (2022). Covering by homothets and illuminating convex bodies. Proceedings of the American Mathematical Society, 150(02), 779-793. https://doi.org/10.1090/proc/15516

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.

Publication Title

Proceedings of the American Mathematical Society

DOI

10.1090/proc/15516