ON THE PERIMETER OF k PAIRWISE DISJOINT CONVEX BODIES CONTAINED IN A CONVEX SET IN THE PLANE

摘要

We prove the following isoperimetric inequality in R-2, conjectured by Glazyrin and MoriAc. Given a convex body S and k pairwise disjoint convex bodies C-1,...,C-k that are contained in S, then Sigma(k)(i=1) Per(C-i) <= Per(S) + 2(k-1)Diam(S). Here Per(.) denotes the perimeter of a set and Diam(.) is the diameter of a set.