Given a planar convex domain , its Cheeger set Ꮿ is defined as the unique minimizer of |∂ X|/|X| among all nonempty open and simply connected subsets X of . We prove an interesting geometric property of Ꮿ , characterize domains which coincide with Ꮿ and provide a constructive algorithm for the determination of Ꮿ .

We present numerical methods to solve optimization problems on the space of convex functions or among convex bodies. Hence convexity is a constraint on the admissible objects, whereas the functionals are not required to be convex. To deal with, our method mix geometrical and numerical algorithms. We give several applications arising from classical problems in geometry and analysis: Alexandrov's problem of finding a convex body of prescribed surface function; Cheeger's problem of a subdomain minimizing the ratio surface area on volume; Newton's problem of the body of minimal resistance. In particular for the latter application, the minimizers are still unknown, except in some particular classes. We give approximate solutions better than the theoretical known ones, hence demonstrating that the minimizers do not belong to these classes.

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We consider the problem of the body of minimal resistance as formulated in [2], Sect. 5: minimize F (u) :where Ω is the unit disc of R 2 , in the class of radial functions u : Ω → [0, M] satisfying a geometrical property (1), corresponding to a single-impact assumption (M > 0 is a given parameter). We prove the existence of a critical value M * of M . For M ≥ M * , there exist a unique local minimizer of the functional. For M < M * , the set of local minimizers is not compact in H 1 , though they all achieve the same value of the functional.

We describe an algorithm to approximate the minimizer of an elliptic functional in the form Ω j(x, u, ∇u) on the set C of convex functions u in an appropriate functional space X. Such problems arise for instance in mathematical economics [4]. A special case gives the convex envelope u * * 0 of a given function u 0 . Let (T n ) be any quasiuniform sequence of meshes whose diameter goes to zero, and I n the corresponding affine interpolation operators. We prove that the minimizer over C is the limit of the sequence (u n ), where u n minimizes the functional over I n (C). We give an implementable characterization of I n (C). Then the finite dimensional problem turns out to be a minimization problem with linear constraints.