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On the diminishing process of Bálint Tóth


Authors: Péter Kevei and Viktor Vígh
Journal: Trans. Amer. Math. Soc. 368 (2016), 8823-8848
MSC (2010): Primary 60D05; Secondary 52A22, 60G99
Published electronically: March 2, 2016
MathSciNet review: 3551590
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Abstract: Let $ K$ and $ K_0$ be convex bodies in $ \mathbb{R}^d$, such that $ K$ contains the origin, and define the process $ (K_n, p_n)$, $ n \geq 0$, as follows: let $ p_{n+1}$ be a uniform random point in $ K_n$, and set $ K_{n+1} = K_n \cap (p_{n+1} + K)$. Clearly, $ (K_n)$ is a nested sequence of convex bodies which converge to a non-empty limit object, again a convex body in $ \mathbb{R}^d$. We study this process for $ K$ being a regular simplex, a cube, or a regular convex polygon with an odd number of vertices. We also derive some new results in one dimension for non-uniform distributions.


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Additional Information

Péter Kevei
Affiliation: MTA-SZTE Analysis and Stochastics Research Group, Bolyai Institute, University of Szeged, H-6720, Szeged, Aradi vértanúk tere 1, Hungary
Email: kevei@math.u-szeged.hu

Viktor Vígh
Affiliation: Department of Geometry, Bolyai Institute, University of Szeged, H-6720, Szeged, Aradi vértanúk tere 1, Hungary
Email: vigvik@math.u-szeged.hu

DOI: https://doi.org/10.1090/tran/6620
Received by editor(s): June 25, 2014
Received by editor(s) in revised form: November 5, 2014, and November 18, 2014
Published electronically: March 2, 2016
Article copyright: © Copyright 2016 American Mathematical Society