Transversal numbers of translates of a convex body

Let FF be a family of translates of a fixed convex set M   in RnRn. Let τ(F)τ(F) and ν(F)ν(F) denote the transversal number and the independence number of FF, respectively. We show that ν(F)⩽τ(F)⩽8ν(F)-5ν(F)⩽τ(F)⩽8ν(F)-5 for n=2n=2 and τ(F)⩽2n-1nnν(F)τ(F)⩽2n-1nnν(F) for n⩾3n⩾3. Furthermore, if M   is centrally symmetric convex body in the plane, then ν(F)⩽τ(F)⩽6ν(F)-3ν(F)⩽τ(F)⩽6ν(F)-3.