Let C be a simple closed curve in the plane, of length l enclosing a region, S, of area A. The isoperimetric problem of Greek geometry was to find the maximum value of A for a fixed l, the obvious candidate for a solution being a circle of radius r = l/2 ? and area A = ?r ² = l ² /4 ?. Since under dilation of the plane by a factor ? > 0, every such planar set is mapped to a set with boundary of length ?l and area ? ² A, as a function of l the maximum value of A will be A ( l) = ?l ² for some constant ? ? 1 /4 ?. Thus A ( l) must be monotone increasing. Minimizing l for fixed A is an equivalent form of the isoperimetric problem whose solution, , is also monotone increasing. The region S may be assumed convex since, if not, its convex closure would only decrease l and increase A.