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].
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
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Proceedings of the American Mathematical Society