A Helly-type theorem for intersections of orthogonally starshaped sets in ?~d

摘要

LetKbe a finite family of orthogonal polytopes in ?~d such that, for every nonempty subfamily K' of K, ∩{K: K in K'}, if nonempty, is a finite union of boxes whose intersection graph is a tree. Assume that every d + 1 (not necessarily distinct) members of K meet in a (nonempty) staircase starshaped set. Then S ≡ ∩{K: K in K} is nonempty and staircase starshaped.