Some relations among volume, intrinsic perimeter and one-dimensional restrictions of BV functions in Carnot groups

Abstract

Let G be a k-step Carnot group. The first aim of this paper is to show an interplay between volume and G-perimeter, using one-dimensional horizontal slicing. What we prove is a kind of Fubini theorem for G-regular submanifolds of codimension one. We then give some applications of this result: slicing of B VG functions, integral geometric formulae for volume and G-perimeter and, making use of a suitable notion of convexity, called G-convexity, we state a Cauchy type formula for G-convex sets. Finally, in the last section we prove a sub-Riemannian Santaló formula showing some related applications. In particular we find two lower bounds for the first eigenvalue of the Dirichlet problem for the Carnot sub-Laplacian ΔG on smooth domains.