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},
mrnumber = {2165404},
zbl = {1150.49022},
language = {en},
url = {http://archive.numdam.org/item/ASNSP_2005_5_4_1_79_0/}
}
TY - JOUR AU - Montefalcone, Francescopaolo TI - Some relations among volume, intrinsic perimeter and one-dimensional restrictions of $BV$ functions in Carnot groups JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2005 SP - 79 EP - 128 VL - 4 IS - 1 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2005_5_4_1_79_0/ LA - en ID - ASNSP_2005_5_4_1_79_0 ER -
%0 Journal Article %A Montefalcone, Francescopaolo %T Some relations among volume, intrinsic perimeter and one-dimensional restrictions of $BV$ functions in Carnot groups %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2005 %P 79-128 %V 4 %N 1 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2005_5_4_1_79_0/ %G en %F 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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