Visibility in crowds of translates of a centrally symmetric convex body

Let BpBp, BqBq be disjoint translates of a centrally symmetric convex body BB in RnRn. A translate BrBr of BB lies between BpBp and BqBq if it overlaps none of BpBp and BqBq and there are points x∈Bpx∈Bp, y∈Bqy∈Bq such that the segment [x,y][x,y] meets BrBr. For a family FF of translates of BB lying between BpBp and BqBq and forming a packing, these two bodies are said to be visible from each other in the system {Bp,Bq}∪F{Bp,Bq}∪F if there exist points x,yx,y like above such that [x,y][x,y] intersects no translate of FF; otherwise BpBp and BqBq are concealed from each other by FF. The concealment number of a Minkowski space with unit ball BB is the infimum of λ>0λ>0 with ‖p−q‖>λ‖p−q‖>λ implying that Bp,BqBp,Bq can be concealed from each other by translates of BB. Continuing the investigations of other authors, we prove several results about concealment numbers of Minkowski planes.