On integration by parts formula on open convex sets in Wiener spaces

In Euclidean space, it is well known that any integration by parts formula for a set of finite perimeter Ω is expressed by the integration with respect to a measure P(Ω , ·) which is equivalent to the one-codimensional Hausdorff measure restricted to the reduced boundary of Ω. The same result has been proved in an abstract Wiener space, typically an infinite-dimensional space, where the surface measure considered is the one-codimensional spherical Hausdorff–Gauss measure S∞-1 restricted to the measure-theoretic boundary of Ω. In this paper, we consider an open convex set Ω and we provide an explicit formula for the density of P(Ω , ·) with respect to S∞-1. In particular, the density can be written in terms of the Minkowski functional p of Ω with respect to an inner point of Ω. As a consequence, we obtain an integration by parts formula for open convex sets in Wiener spaces.