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 $\mathrm{\Omega }$ is expressed by the integration with respect to a measure $P(\mathrm{\Omega },\cdot )$ which is equivalent to the one-codimensional Hausdorff measure restricted to the reduced boundary of $\mathrm{\Omega }$. 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 $\mathscr S^{\infty-1}$restrictedtothemeasure-theoreticboundaryof$\Omega$. In this paper we consider an open convex set $\mathrm{\Omega }$ and we provide an explicit formula for the density of $P(\mathrm{\Omega },\cdot )$ with respect to $\mathscr S^{\infty-1}$. In particular, the density can be written in terms of the Minkowski functional $\text{p}$ of $\mathrm{\Omega }$ with respect to an inner point of $\mathrm{\Omega }$. As a consequence, we obtain an integration by parts formula for open convex sets in Wiener spaces.