On the total perimeter of homothetic convex bodies in a convex container

摘要

For two planar convex bodies, C and D, consider a packing S of n positive homothets of C contained in D. We estimate the total perimeter of the bodies in S, denoted per(S), in terms of per(D) and n.When all homothets ofC touch the boundary of the container D, we show that either per(S) = O(log n) or per(S) = O(1), depending on how C and D "fit together". Apart from the constant factors, these bounds are the best possible. Specifically, we prove that per(S) = O(1) if D is a convex polygon and every side of D is parallel to a corresponding segment on the boundary of C (for short, D is parallel to C) and per(S) = O(log n) otherwise.When D is parallel to C but the homothets of C may lie anywhere in D, we show that per(S) = O((1+esc(S)) log n/ log log n), where esc(S) denotes the total distance of the bodies in S from the boundary of D. Apart from the constant factor, this bound is also the best possible.