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

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 of C touch the boundary of the container D , we show that either per(S)=O(logn) 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(logn) 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))logn/loglogn) , 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.

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