Hostname: page-component-7dd5485656-j7khd Total loading time: 0 Render date: 2025-10-31T06:13:43.439Z Has data issue: false hasContentIssue false
  • English
  • Français

Random Symmetrizations of Convex Bodies

Part of: Geometric probability and stochastic geometry General convexity

Published online by Cambridge University Press:  22 February 2016

D. Coupier  and
Yu. Davydov
D. Coupier*
Affiliation:
Université Lille 1
Yu. Davydov*
Affiliation:
Université Lille 1
*
∗ Postal address: Laboratoire Paul Painlevé, UMR CNRS 8524, Université Lille 1, 59 655 Villeneuve d'Ascq Cédex, France.
∗∗ Email address: david.coupier@math.univ-lille1.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we investigate the asymptotic behavior of sequences of successive Steiner and Minkowski symmetrizations. We state an equivalence result between the convergences of those sequences for Minkowski and Steiner symmetrizations. Moreover, in the case of independent (and not necessarily identically distributed) directions, we prove the almost-sure convergence of successive symmetrizations at exponential rate for Minkowski, and at rate with c > 0 for Steiner.

Keywords

Stochastic geometryconvex geometrylimit shapeSteiner and Minkowski symmetrizations

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 52A22: Random convex sets and integral geometry

Information

Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 46 , Issue 3 , September 2014 , pp. 603 - 621
Copyright
© Applied Probability Trust 

References

Bianchi, G., Burchard, A., Gronchi, P. and Volčič, A. (2012). Convergence in shape of Steiner symmetrizations. Indiana Univ. Math. J. 61, 1695–1710.CrossRefGoogle Scholar
Bianchi, G. et al. (2011). A countable set of directions is sufficient for Steiner symmetrization. Adv. Appl. Math. 47, 869–873.CrossRefGoogle Scholar
Bokowski, J. and Heil, E. (1986). Integral representations of quermassintegrals and Bonnesen-style inequalities. Arch. Math. (Basel) 47, 79–89.Google Scholar
Burchard, A. and Fortier, M. (2013). Random polarizations. Adv. Math. 234, 550–573.CrossRefGoogle Scholar
Groemer, H. (2000). On the symmetric difference metric for convex bodies. Beiträge Algebra Geom. 41, 107–114.Google Scholar
Gruber, P. M. (2007). Convex and Discrete Geometry (Fundamental Principles Math. Sci. 336). Springer, Berlin.Google Scholar
Klain, D. A. (2012). Steiner symmetrization using a finite set of directions. Adv. Appl. Math. 48, 340–353.Google Scholar
Klartag, B. (2004). Rate of convergence of geometric symmetrizations. Geom. Funct. Anal. 14, 1322–1338.CrossRefGoogle Scholar
Mani-Levitska, P. (1986). Random Steiner symmetrizations. Studia Sci. Math. Hungar. 21, 373–378.Google Scholar
Meyn, S. and Tweedie, R. L. (2009). Markov Chains and Stochastic Stability, 2nd edn. Cambridge University Press.CrossRefGoogle Scholar
Müller, C. (1966). Spherical Harmonics (Lecture Notes Math. 17). Springer, Berlin.CrossRefGoogle Scholar
Schneider, R. (1993). Convex Bodies: The Brunn–Minkowski Theory (Encyclopedia Math. Appl. 44). Cambridge University Press.Google Scholar
Volčič, A. (2013). Random Steiner symmetrizations of sets and functions. Calc. Var. Partial Differential Equat. 46, 555–569.Google Scholar