COVERING by HOMOTHETS and ILLUMINATING CONVEX BODIES

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, g(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. h(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 g(B) ≤ h(B), find upper and lower bounds for both numbers, and discuss several general conjectures. In particular, we show that h(B) > 2− 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].