Small-Ball Probabilities for the Volume of Random Convex Sets
- PDF / 459,901 Bytes
- 46 Pages / 439.37 x 666.142 pts Page_size
- 75 Downloads / 182 Views
Small-Ball Probabilities for the Volume of Random Convex Sets Grigoris Paouris · Peter Pivovarov
Received: 17 May 2012 / Revised: 30 December 2012 / Accepted: 11 February 2013 / Published online: 26 March 2013 © Springer Science+Business Media New York 2013
Abstract We prove small-deviation estimates for the volume of random convex sets. The focus is on convex hulls and Minkowski sums of line segments generated by independent random points. The random models considered include (Lebesgue) absolutely continuous probability measures with bounded densities and the class of log-concave measures. Keywords Volume of random sets · Small deviations · Rearrangements · Affine quermassintegrals
1 Introduction The focus of this paper is distributional inequalities for the volume of random convex sets. Typical models involve convex hulls or Minkowski sums of line segments generated by independent random points in Rn . Specifically, let μ be a probability measure on Rn . Sample N ≥ n independent points X 1 , . . . , X N according to μ. Let K N be the absolute convex hull of the X i ’s, i.e., K N := conv ± X 1 , . . . , ±X N
(1.1)
G. Paouris Department of Mathematics, Texas A&M University, Mailstop 3368, College Station, TX, USA e-mail: [email protected] P. Pivovarov (B) Mathematics Department, University of Missouri, Columbia, MO 65211, USA e-mail: [email protected]
123
602
Discrete Comput Geom (2013) 49:601–646
and let Z N be the zonotope, i.e., the Minkowski sum of the line segments [−X i , X i ], Z N :=
N i=1
[−X i , X i ] =
N
λi X i : λi ∈ [−1, 1], i = 1, . . . , N .
(1.2)
i=1
The literature contains a wealth of results aimed at quantifying the size of K N and its non-symmetric analogue conv X 1 , . . . , X N in terms of metric quantities such as volume, surface area and mean-width; especially in the asymptotic setting where the dimension n is fixed and N → ∞. The measure μ strongly determines the corresponding properties of K N and Z N . Common models include the case when μ is the standard Gaussian measure, see e.g., [10,39]; the uniform measure on a convex body, see e.g., the survey [7]; among many others, e.g., [71]. These are just a sample of recent articles and we refer the reader to the thorough list of references given therein. A different asymptotic setting involves the case when the dimension n is large and one is interested in precise dependence on N and phenomena that hold uniformly for a large family of measures μ. In this setting, various geometric properties of K N and Z N such as Banach–Mazur distance, in-radius and other metric quantities have been analyzed. For zonotopes, see e.g., [14,15,35]. Concerning K N there have been a number of recent results with special attention paid to estimates that hold “with high probability.” These include, for instance, the case when μ is the uniform measure on the vertices of the cube [29], measures with “Gaussian-like” features [45,50] and the case when μ is the uniform measure on a convex body [21,30]. We are interested in dist
Data Loading...
