Perfect partitions of convex sets in the plane

For a region X in the plane, wedenoteby area(X) the areaof X and by l(partial derivative(X)) the length of the boundary of X. Let S be a convex set in the plane, let n greater than or equal to 2 be an integer, and let alpha(1), alpha(2),...., alpha(n) be positive real numbers such that alpha(1) + alpha(2) +(...)+ alpha(n) = 1 and 0 < alpha(i) less than or equal to (1)/(2) for all 1 less than or equal to i less than or equal to n. then we shall show that S can be partitioned into n disjoint convex subsets T-1, T-2,..., T-n so that each T-i satisfies the following three conditions: (i) area(T-i) = alpha(i) x area(S); (ii) l(T-i boolean AND partial derivative(S)) = alpha(i) x (partial derivative(S)); and (iii) T-i boolean AND partial derivative(S) consists of exactly one continuous curve.