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

在欧几里德空间中，这是众所周知的，对于一组有限的周长任何集成由部件式\（\欧米茄\）通过积分相对于度量表示\（P（\欧米茄，\ CDOT）\） ，其是等效于限制于\（\ Omega \）的缩减边界的一维Hausdorff测度。在抽象的维纳空间（通常是无限维空间）中已证明了相同的结果，其中考虑的表面度量是一维球形Hausdorff–Gauss度量\（\ mathscr {S} ^ {\ infty -1} \）限于\（\ Omega \）的量度理论边界。在本文中，我们考虑开放凸集\（\ Omega \）并针对\（\ mathscr {S} ^ {\ infty -1} \）给出\（P（\ Omega，\ cdot）\）密度的显式公式。特别地，该密度可以在闵可夫斯基功能方面被写入\（\ mathfrak {P} \）的\（\欧米茄\）相对于的内点\（\欧米茄\） 。结果，我们获得了维纳空间中开放凸集的零件公式积分。