Issue |
ESAIM: PS
Volume 18, 2014
|
|
---|---|---|
Page(s) | 854 - 880 | |
DOI | https://doi.org/10.1051/ps/2014008 | |
Published online | 29 October 2014 |
Compact convex sets of the plane and probability theory∗
CNRS, LaBRI, Université de Bordeaux, 351 cours de la Libération, 33405
Talence cedex, France
name@labri.fr; renault@labri.fr
Received:
11
December
2012
Revised:
15
July
2013
The Gauss−Minkowski
correspondence in ℝ2 states the existence of a homeomorphism between the
probability measures μ on [0,2π] such that and the compact convex sets (CCS) of the plane with
perimeter 1. In this article, we bring out explicit formulas relating the border of a CCS
to its probability measure. As a consequence, we show that some natural operations on CCS
– for example, the Minkowski sum – have natural translations in terms of probability
measure operations, and reciprocally, the convolution of measures translates into a new
notion of convolution of CCS. Additionally, we give a proof that a polygonal curve
associated with a sample of n random variables (satisfying
) converges to a CCS associated with μ at speed √n, a result much similar to the convergence of the
empirical process in statistics. Finally, we employ this correspondence to present models
of smooth random CCS and simulations.
Mathematics Subject Classification: 52A10 / 60B05 / 60D05 / 60F17 / 60G99
Key words: Random convex sets / symmetrisation / weak convergence / Minkowski sum
© EDP Sciences, SMAI 2014
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