Perimeter of sublevel sets in infinite dimensional spaces

TL;DR

This paper investigates the perimeter measure in infinite-dimensional spaces, comparing it with the Airault-Malliavin surface measure, and establishes conditions under which convex sets have finite perimeter.

Contribution

It demonstrates that all open convex subsets of abstract Wiener spaces have finite perimeter, contrasting with compact convex domains which may not.

Findings

Open convex sets in Wiener spaces have finite perimeter.

Counter-example shows compact convex domains can have infinite perimeter.

Comparison between perimeter measure and Airault-Malliavin surface measure.

Abstract

We compare the perimeter measure with the Airault-Malliavin surface measure and we prove that all open convex subsets of abstract Wiener spaces have finite perimeter. By an explicit counter-example, we show that in general this is not true for compact convex domains.