The aim of this paper is to study the isoperimetric problem with fixed volume inside convex sets and other related geometric variational problems in the Gauss space, in both the finite and infinite dimensional case. We first study the finite dimensional case, proving the existence of a maximal Cheeger set which is convex inside any bounded convex set. We also prove the uniqueness and convexity of solutions of the isoperimetric problem with fixed volume inside any convex set. Then we extend these results in the context of the abstract Wiener space, and for that we study the total variation denoising problem in this context.

V. Caselles and M. Novaga acknowledge partial support by Acción Integrada Hispano Italiana HI2008-0074. V. Caselles also acknowledges by MICINN project, reference MTM2009-08171, by GRC reference 2009 SGR 773 and by “ICREA Acadèmia” for excellence in research, the last two funded by the Generalitat de Catalunya. M. Miranda acknowledges partial support by the GNAMPA project “Metodi geometrici per analisi in spazi non Euclidei; spazi metrici doubling, gruppi di Carnot e spazi di Wiener”. M. Novaga acknowledges partial support by the Research Institute “Le STUDIUM”.