Covering a convex body by its negative homothetic copies. (Q5930556)

Let \(f_\lambda(C)\) be the smallest number of homothetic copies of a fixed negative ratio \(\lambda\) of a convex body \(C\in{\mathbb R}^d\) sufficient to cover \(C\). The main results are: (1) If \(-d\leq\lambda\leq-d/2\), then \(f_\lambda(C)\leq 2^{\lceil2d+2\lambda\rceil}\). (2) If \(-d/2\leq\lambda<0\), then \(f_\lambda(C)\leq \lceil -d/\lambda\rceil^d\). (3) If \(\lambda=-1+d^{-d}(d+1)^{-1}\), then \(f_\lambda(C)\leq d^d+1\). (4) If \(d=2\) then \(f_{-4/3}(C)\leq 2\). (Here the ratio \(-4/3\) is best possible for two copies.)