On coverings of convex sets by translates of slabs

Abstract:

Let $({S_1},{S_2}, \ldots )$ be a sequence of slabs in euclidean $n$-dimensional space and let ${t_i}$ denote the thickness of ${S_i}$. It is shown that the condition $\sum {t_i^{(n + 1)/2} = \infty }$ implies that every convex set can be covered by translates of the slabs ${S_i}$, and that the exponent $(n + 1)/2$ is, in a certain sense, best possible.