The paper is motivated by Hadwiger’s covering conjecture, which claims that every \(n\)-dimensional convex body can be covered by at most \(2^n\) smaller homothetic copies of itself. Recent work on Hadwiger’s conjecture follows the quantitative program layed out in [C. Zong, Sci. China, Math. 53, No. 9, 2551–2560 (2010; Zbl 1237.52014)], which in particular involves studying the covering functionals \(\Gamma_m(K)\) for various classes of convex bodies \(K\). For any positive integer \(m\) the functional \(\Gamma_m(K)\) is defined as the smallest \(\gamma > 0\) such that \(K\) can be covered by at most \(m\) translates of \(\gamma K\).
The main result of the authors concerns the class of zonotopes and establishes the bound \(\Gamma_{2^n}(Z) \leq \frac{1 + 2k}{2 + 2k}\), for every \(n\)-dimensional zonotope \(Z\) that can be written as the Minkowski sum of \(n+k\) line segments. This result is obtained by estimating, more generally, the covering functionals of the convex hull of the union of \(p\) convex bodies by the covering functionals of the \(p\) constituents.