# hw8sol - EE364b Prof. S. Boyd EE364b Homework 8 Solution 1....

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EE364bProf. S. BoydEE364b Homework 8 Solution1.Circle packing.The goal is to placeNpoints in a unit box inR2so as to maximizethe minimum distance between any of the points:maximizeDsubject tokxi-xjk2D,i6=j0xi1,(1)with variablesx1, . . . , xNR2andDR. This problem, and various variations on it,are sometimes called circle-packing problems, since they come down to placing circlesin such a way that they don’t intersect (except at the boundaries). You can probablyguess what good solutions look like.(a)Area bound.Derive an upper boundDon the optimal value, using the followingargument.ForxiandDfeasible, theNdisks centered atxiwith diameterDdon’t overlap, except on their boundaries; these disks also lie inside the box{x| -(D/2)1x(1 +D/2)1}. Therefore, the total area of the disks is notmore than the area of this box.

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