Representation, relaxation and convexity for variational problems in Wiener spaces

The aim of this paper is to study the convexity of the minimizers of some variational problems in Wiener spaces. In the Euclidean setting convexity is a widely discussed issue [25]. Recently, following previous work by Korevaar [26] and Alvarez, Lasry and Lions [2], Alter, Caselles and Chambolle [1, 12] showed the convexity of solutions to variational problems involving functionals with linear growth and in particular to the prescribed curvature problem. Using quite different techniques, Figalli and Maggi [19] proved the convexity of small mass minimizers of this problem. The main goal of this paper, is to extend these results to the (finite dimensional) Gauss space and to the (infinite dimensional) Wiener space. In this setting, very few results are currently available. To the best of our knowledge, the only result in this direction is contained in [13], where the authors proved the convexity of the solutions of the isoperimetric problem in convex domains. More explicitly they prove the following: