Let $A$ be a $C^2$ curve of length $L(A)$ in some Euclidean space. Let $P_n$ be a sequence of randomly chosen polygons with $n$ vertices which are inscribed in $A$. It is shown that with probability 1 $\lim n^2\lbrack L(A) - L(P_n)\rbrack = \frac{1}{4} \int_A \kappa^2(s) ds$ where $\kappa$ is the curvature.
Publié le : 1982-08-14
Classification:
Random polygons,
Random partitions of an internal,
60F15,
53A05
@article{1176993798,
author = {Drobot, Vladimir},
title = {Probabilistic Version of a Curvature Formula},
journal = {Ann. Probab.},
volume = {10},
number = {4},
year = {1982},
pages = { 860-862},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993798}
}
Drobot, Vladimir. Probabilistic Version of a Curvature Formula. Ann. Probab., Tome 10 (1982) no. 4, pp. 860-862. http://gdmltest.u-ga.fr/item/1176993798/