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Variance estimates for random disc-polygons in smooth convex discs

Part of: Geometric probability and stochastic geometry General convexity

Published online by Cambridge University Press:  16 January 2019

Ferenc Fodor  and
Viktor Vígh
Ferenc Fodor*
Affiliation:
University of Szeged
Viktor Vígh*
Affiliation:
University of Szeged
*
* Postal address: Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, 6720 Szeged, Hungary.
* Postal address: Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, 6720 Szeged, Hungary.
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Abstract

In this paper we prove asymptotic upper bounds on the variance of the number of vertices and the missed area of inscribed random disc-polygons in smooth convex discs whose boundary is C+2. We also consider a circumscribed variant of this probability model in which the convex disc is approximated by the intersection of random circles.

Keywords

Disc-polygonrandom approximationvariance

MSC classification

Primary: 52A22: Random convex sets and integral geometry
Secondary: 60D05: Geometric probability and stochastic geometry
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 4 , December 2018 , pp. 1143 - 1157
Copyright
Copyright © Applied Probability Trust 2018 

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References

[1]Bárány, I. (2008). Random points and lattice points in convex bodies. Bull. Amer. Math. Soc. 45, 339–365.Google Scholar
[2]Bárány, I. and Reitzner, M. (2010). On the variance of random polytopes. Adv. Math. 225, 1986–2001.Google Scholar
[3]Bárány, I. and Steiger, W. H. (2013). On the variance of random polygons. Comput. Geom. 46, 173–180.Google Scholar
[4]Bárány, I. and Vu, V. H. (2007). Central limit theorems for Gaussian polytopes. Ann. Prob. 35, 1593–1621.Google Scholar
[5]Bárány, I., Fodor, F. and Vígh, V. (2010). Intrinsic volumes of inscribed random polytopes in smooth convex bodies. Adv. Appl. Prob. 42, 605–619.Google Scholar
[6]Bárány, I., Hug, D., Reitzner, M. and Schneider, R. (2017). Random points in halfspheres. Random Structures Algorithms 50, 3–22.Google Scholar
[7]Bezdek, K. (2018). From r-dual sets to uniform contractions. Aequationes Math. 92, 123–134.Google Scholar
[8]Bezdek, K. and Connelly, R. (2002). Pushing disks apart—the Kneser-Poulsen conjecture in the plane. J. Reine Angew. Math. 553, 221–236.Google Scholar
[9]Bezdek, K. and Naszódi, O. (2018). The Kneser–Poulsen conjecture for special contractions. Discrete Comput. Geom. 60, 967–980.Google Scholar
[10]Bezdek, K., Lángi, Z., Naszódi, M. and Papez, P. (2007). Ball-polyhedra. Discrete Comput. Geom. 38, 201–230.Google Scholar
[11]Böröczky, K. J. and Schneider, R. (2010). The mean width of circumscribed random polytopes. Canad. Math. Bull. 53, 614–628.Google Scholar
[12]Böröczky, K. J., Fodor, F. and Hug, D. (2010). The mean width of random polytopes circumscribed around a convex body. J. Lond. Math. Soc. 81, 499–523.Google Scholar
[13]Böröczky, K. J., Fodor, F., Reitzner, M. and Vígh, V. (2009). Mean width of random polytopes in a reasonably smooth convex body. J. Multivariate Anal. 100, 2287–2295.Google Scholar
[14]Efron, B. and Stein, C. (1981). The jackknife estimate of variance. Ann. Statist. 9, 586–596.Google Scholar
[15]Fejes T’oth, G. and Fodor, F. (2015). Dowker-type theorems for hyperconvex discs. Period. Math. Hungar. 70, 131–144.Google Scholar
[16]Fodor, F. and Vígh, V. (2012). Disc-polygonal approximations of planar spindle convex sets. Acta Sci. Math. 78, 331–350.Google Scholar
[17]Fodor, F., Hug, D. and Ziebarth, I. (2016). The volume of random polytopes circumscribed around a convex body. Mathematika 62, 283–306.Google Scholar
[18]Fodor, F., Kevei, P. and Vígh, V. (2014). On random disc polygons in smooth convex discs. Adv. Appl. Prob. 46, 899–918.Google Scholar
[19]Fodor, F., Kurusa, Á. and Vígh, V. (2016). Inequalities for hyperconvex sets. Adv. Geom. 16, 337–348.Google Scholar
[20]Jahn, T., Martini, H. and Richter, C. (2017). Ball convex bodies in Minkowski spaces. Pacific J. Math. 289, 287–316.Google Scholar
[21]Mayer, A. E. (1935). Eine Überkonvexität. Math. Z. 39, 511–531.Google Scholar
[22]Paouris, G. and Pivovarov, P. (2017). Random ball-polyhedra and inequalities for intrinsic volumes. Monatsh. Math. 182, 709–729.Google Scholar
[23]Reitzner, M. (2003). Random polytopes and the Efron-Stein jackknife inequality. Ann. Prob. 31, 2136–2166.Google Scholar
[24]Reitzner, M. (2005). Central limit theorems for random polytopes. Prob. Theory Relat. Fields 133, 483–507.Google Scholar
[25]Rényi, A. and Sulanke, R. (1963). Über die konvexe Hülle von n zufällig gewählten Punkten. Z. Wahrscheinlichkeitsth. 2, 75–84.Google Scholar
[26]Rényi, A. and Sulanke, R. (1963). Über die konvexe Hülle von n zufällig gewählten Punkten, II. Z. Wahrscheinlichkeitsth. 3, 138–147.Google Scholar
[27]Santaló, L. A. (1946). On plane hyperconvex figures. Summa Brasil. Math. 1, 221–239.Google Scholar
[28]Schneider, R. (2008). Recent results on random polytopes. Boll. Unione Mat. Ital. 1, 17–39.Google Scholar
[29]Schneider, R. (2014). Convex Bodies: The Brunn-Minkowski Theory, 2nd end. Cambridge University Press.Google Scholar
[30]Schneider, R. (2017). Discrete aspects of stochastic geometry. In Handbook of Discrete and Computational Geometry, 3rd edn. CRC Press, Boca Raton, FL, pp. 299–329.Google Scholar
[31]Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.Google Scholar
[32]Schreiber, T. and Yukich, J. E. (2008). Variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points. Ann. Prob. 36, 363–396.Google Scholar
[33]Thäle, C., Turchi, N. and Wespi, F. (2018). Random polytopes: central limit theorems for intrinsic volumes. Proc. Amer. Math. Soc. 146, 3063–3071.Google Scholar
[34]Turchi, N. and Wespi, F. (2018). Limit theorems for random polytopes with vertices on convex surfaces. Adv. Appl. Prob, 50, 1227–1245.Google Scholar
[35]Vu, V. H. (2005). Sharp concentration of random polytopes. Geom. Funct. Anal. 15, 1284–1318.Google Scholar
[36]Vu, V. H. (2006). Central limit theorems for random polytopes in a smooth convex set. Adv. Math. 207, 221–243.Google Scholar
[37]Weil, W. and Wieacker, J. A. (1993). Stochastic geometry. In Handbook of Convex Geometria, Vol B, North-Holland, Amsterdam, pp. 1391–1438.Google Scholar