Volume 18, 2014
|Page(s)||854 - 880|
|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
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
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.