Reference no: EM132399156

Consider a set {s1,s2,...,sn} of n squares of arbitrary size all centered on the x-axis. Let S be the boundary of the union of the n squares. Let lp be the leftmost point and rp be the rightmost point of the intersection of S with the x-axis. Show that the length of S is at most 4 times the length of the segment [lp, rp].

Hint: use induction. Notice that these hold easily if the set consists of only one square. For the inductive step, you have to analyse what happens when you have more than one square. Also prove that the length of S is at most π when you replace squares with circles centred on the x-axis.