Let be a -step Carnot group. The first aim of this paper is to show an interplay between volume and -perimeter, using one-dimensional horizontal slicing. What we prove is a kind of Fubini theorem for -regular submanifolds of codimension one. We then give some applications of this result: slicing of functions, integral geometric formulae for volume and -perimeter and, making use of a suitable notion of convexity, called -convexity, we state a Cauchy type formula for -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 on smooth domains.
@article{ASNSP_2005_5_4_1_79_0, author = {Montefalcone, Francescopaolo}, title = {Some relations among volume, intrinsic perimeter and one-dimensional restrictions of $BV$ functions in Carnot groups}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {79--128}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 4}, number = {1}, year = {2005}, zbl = {1150.49022}, mrnumber = {2165404}, language = {en}, url = {archive.numdam.org/item/ASNSP_2005_5_4_1_79_0/} }
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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