Covering cross-polytopes with smaller homothetic copies

Abstract

Let $ C_{n} $ be an $ n $-dimensional cross-polytope and $ \Gamma_{p}(C_{n}) $ be the smallest positive number $ \gamma $ such that $ C_{n} $ can be covered by $ p $ translates of $ \gamma C_{n} $. We obtain better estimates of $ \Gamma_{2^n}(C_n) $ for small $ n $ and a universal upper bound of $ \Gamma_{2^n}(C_n) $ for all positive integers $ n $.

Keywords

• convex body
• covering functional
• hadwiger's covering conjecture
• homothetic copy