Blocking numbers and fixing numbers of convex bodies (Q1045195)

The blocking number of a convex body \(K\) in \(E^d\) is the minimum number of non-overlapping translates of \(K\) that touch \(K\) at its boundary and prevent any other translate from touching \(K\). In this paper it is proved that the blocking number of the crosspolytope in \(E^3\) is 6 and that the blocking number of the \(l_p\) unit ball in \(E^3\) is at most 6 for \(\frac{\ln 3}{\ln 2} < p <\infty\). There is also given a lower and an upper bound for the blocking number of a \(d\)-dimensional cylinder whose base is a \((d-1)\)-dimensional convex body. Moreover, the fixing number of a convex body \(K\) is introduced. This number is defined to be the minimum number of non-overlapping translates of \(K\) that touch \(K\) at its boundary and fix \(K\). A lower and an upper bound for this parameter for an arbitrary \(d\)-dimensional convex body are presented which cannot be improved in general.