The Annals of Probability

Probabilistic Version of a Curvature Formula

Vladimir Drobot

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Abstract

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.

Article information

Source
Ann. Probab., Volume 10, Number 3 (1982), 860-862.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993798

Digital Object Identifier
doi:10.1214/aop/1176993798

Mathematical Reviews number (MathSciNet)
MR659557

Zentralblatt MATH identifier
0486.60016

JSTOR
links.jstor.org

Subjects
Primary: 60F15: Strong theorems
Secondary: 53A05: Surfaces in Euclidean space

Keywords
Random polygons Random partitions of an internal

Citation

Drobot, Vladimir. Probabilistic Version of a Curvature Formula. Ann. Probab. 10 (1982), no. 3, 860--862. doi:10.1214/aop/1176993798. https://projecteuclid.org/euclid.aop/1176993798


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