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On the Hausdorff distance between a convex set and an interior random convex hull

Part of: Geometric probability and stochastic geometry General convexity

Published online by Cambridge University Press:  01 July 2016

H. Bräker ,
T. Hsing  and
N. H. Bingham
H. Bräker*
Affiliation:
University of Bern
T. Hsing*
Affiliation:
Singapore National University
N. H. Bingham*
Affiliation:
Birkbeck College (Univ. London)
*
∗ Postal address: University of Bern, 3012 Bern, Switzerland.
∗∗ Postal address: Singapore National University, 10 Kent Ridge Crescent, Singapore 119260K.
∗∗∗ Birkbeck College (Univ. London), London WC1E 7HX, UK. Email address: n.bingham@statistics.bbk.ac.uk
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Abstract

The problem of estimating an unknown compact convex set K in the plane, from a sample (X1,···,Xn) of points independently and uniformly distributed over K, is considered. Let Kn be the convex hull of the sample, Δ be the Hausdorff distance, and Δn := Δ (K, Kn). Under mild conditions, limit laws for Δn are obtained. We find sequences (an), (bn) such that

(Δn - bn)/an → Λ (n → ∞), where Λ is the Gumbel (double-exponential) law from extreme-value theory. As expected, the directions of maximum curvature play a decisive role. Our results apply, for instance, to discs and to the interiors of ellipses, although for eccentricity e < 1 the first case cannot be obtained from the second by continuity. The polygonal case is also considered.

Keywords

Convex setconvex hullHausdorff metriclimit lawGumbel distributionextreme value theorysmooth boundarypolygonmoving boundaryhome range

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 52A22: Random convex sets and integral geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 30 , Issue 2 , June 1998 , pp. 295 - 316
Copyright
Copyright © Applied Probability Trust 1998 

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