Perfect Partitions of Convex Sets in the Plane

For a region X in the plane, we denote by area(X) the area of X and by ℓ (∂ (X)) the length of the boundary of X . Let S be a convex set in the plane, let n ≥ 2 be an integer, and let α1, α2, . . . ,αn be positive real numbers such that α1+α2+ ⋅ ⋅ ⋅ +αn=1 and 0< αi ≤ 1/2 for all 1 ≤ i ≤ n . Then we shall show that S can be partitioned into n disjoint convex subsets T1, T2, . . . ,Tn so that each Ti satisfies the following three conditions: (i) area(Ti)=αi × area(S) ; (ii) ℓ (Ti ∩ ∂ (S))= αi × ℓ (∂ (S)) ; and (iii) Ti ∩ ∂ (S) consists of exactly one continuous curve.