On a volume constrained variational problem

W (∇u) dx where LN ({u = zi}) = αi, i = 1,..., P, is proved for the case in which zi are extremal points of a compact, convex set in Rd and under suitable assumptions on a class of quasiconvex energy densities W. Optimality properties are studied in the scalar-valued problem where d = 1, P = 2, W (ξ) = |ξ|2, and the Γ-limit as the sum of the measures of the 2 phases tends to LN (Ω) is identified. Minimizers are fully characterized when N = 1, and candidates for solutions are studied for the circle and the square in the plane