Local Measure of Convex Surfaces induced by the Wiener Measure of Paths

The Wiener measure induces a measure of closed, convex, (d-1)-dimensional, Euclidean (hyper-)surfaces that are the convex hulls of closed d-dimensional Brownian bridges. I present arguments and numerical evidence that this measure, for odd d, is generated by a local classical action of length dimension two that depends on geometric invariants of the (d-1)-dimensional surface only.