Covering functionals of convex polytopes with few vertices (Q2161400)

Motivated by Hadwiger's covering conjecture, the authors investigate the covering functional of convex bodies. This functional takes a natural number \(m\) and a convex body \(K\) as inputs, and is defined as the minimum dilation factor \(\gamma\) such that the convex body is the subset of the union of \(m\) translates of \(\gamma K\). In Section 3, the covering functional of \(\ell_p\)-balls in \(\mathbb{R}^n\) and of their portions in the nonnegative orthant are studied. This is done by providing explicit unions of dilated translates covering the original bodies, involving lattice points in crosspolytopes. In Theorem 8, it is shown that the covering functional of the \(\ell_1\)-ball in \(\mathbb{R}^m\) and its portion in the nonnegative orthant are upper bounds for the covering functional of convex polytopes with \(m+1\) vertices and centrally symmetric polytopes with \(2m\) vertices, respectively. The main theorem of this paper (Theorem 9) then gives upper bounds of the covering functional at \(2^n\) of polytopes with \(m\geq n+1\) vertices and centrally symmetric polytopes with \(2m\geq 2n\) vertices, purely in terms of \(n\) and \(m\).