Some relations among volume, intrinsic perimeter and one-dimensional restrictions of BV functions in Carnot groups
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 1, pp. 79-128.

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 BV 𝔾 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 Δ 𝔾 on smooth domains.

Classification : 49Q15, 46E35, 22E60
Montefalcone, Francescopaolo 1

1 Dipartimento di Matematica Università degli Studi di Bologna Piazza di P. ta S. Donato, 5 40126 Bologna, Italia
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Montefalcone, Francescopaolo. Some relations among volume, intrinsic perimeter and one-dimensional restrictions of $BV$ functions in Carnot groups. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 1, pp. 79-128. http://archive.numdam.org/item/ASNSP_2005_5_4_1_79_0/

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