Randomized coverings of a convex body with its homothetic copies, and illumination

TL;DR

This paper introduces a probabilistic model for illuminating convex bodies using independent light sources, improving bounds on the illumination number and parameter, advancing understanding in geometric covering problems.

Contribution

It develops a probabilistic approach to convex body illumination, refining existing bounds and extending previous estimates in the field.

Findings

Recovered C.A. Rogers' upper bounds for the illumination number.

Improved estimates for a generalized illumination parameter.

Enhanced probabilistic methods for convex body covering.

Abstract

We present a probabilistic model of illuminating a convex body by independently distributed light sources. In addition to recovering C.A. Rogers' upper bounds for the illumination number, we improve previous estimates of J. Januszewski and M. Nasz\'odi for a generalized version of the illumination parameter.