For a region X in the plane, we denote by area(X ) the area of X and by (∂(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 α 1 , α 2 , . . . , α n be positive real numbers such that α 1 + α 2 + · · · + α n = 1 and 0 < α i ≤ 1 2 for all 1 ≤ i ≤ 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 ) = α i × area(S); (ii) (T i ∩ ∂(S)) = α i × (∂(S)); and (iii) T i ∩ ∂(S) consists of exactly one continuous curve.