Abstract

Let $\mathbb{G}$ be a $k$-step Carnot group. The first aim of this paper is to show an interplay between volume and $\mathbb{G}$-perimeter, using one-dimensional horizontal slicing. What we prove is a kind of Fubini theorem for $\mathbb{G}$-regular submanifolds of codimension one. We then give some applications of this result: slicing of $BV_{\mathbb{G}}$ functions, integral geometric formulae for volume and $\mathbb{G}$-perimeter and, making use of a suitable notion of convexity, called $\mathbb{G}$-convexity, we state a Cauchy type formula for $\mathbb{G}$-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 $\Delta _{\mathbb{G}}$ on smooth domains