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On the Number of Vertices of the Convex Hull of Random Points in a Square and a Triangle

    Christian Buchta

Sitzungsberichte und Anzeiger der mathematisch-naturwissenschaftlichen Klasse, Jahrgang 2009/10, pp. AII_2009_s3-AII_2009_s10, 2009/03/12

Abteilung I: Biologische Wissenschaften und Erdwissenschaften
Abteilung II: Mathematische, Physikalische und Technische Wissenschaften
143. Band, Jahrgang 2009 – Anzeiger II
218. Band, Jahrgang 2009 – Sitzungsberichte II
213. Band, Jahrgang 2010 – Sitzungsberichte I
219. Band, Jahrgang 2010 – Sitzungsberichte II

doi: 10.1553/SundA2009_2010sAII_2009_s3

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doi:10.1553/SundA2009_2010sAII_2009_s3


Abstract

Assume that n points are chosen independently and according to the uniform distribution from a convex polygon C. Consider the convex hull of the randomly chosen points. The probabilities pk(n) (C) that the convex hull has exactly k vertices are stated for all k in the cases that C is a square (equivalently a parallelogram) or a triangle.

Keywords: Convex_hull random_points random_polygons

REFERENCES [10] SCHNEIDER, R., (1988) Random approximation of convex sets. J. Microscopy 151: 211–227[11] SCHNEIDER, R., (2004) Discrete aspects of stochastic geometry. In: GOODMAN, J. E., O’ROURKE, J. (eds.) Handbook of Discrete and Computational Geometry, 2nd edn., pp. 255–278. Chapman and Hall/CRC, Boca Raton, Florida[12] SCHNEIDER, R., WEIL, W. (2008) Stochastic and Integral Geometry. Springer, Berlin[13] WEIL, W., WIEACKER, J. A. (1993) Stochastic geometry. In: GRUBER, P. M., WILLS, J. M. (eds.) Handbook of Convex Geometry, Vol. B, pp. 1391–1438. North-Holland/Elsevier, Amsterdam[1] AFFENTRANGER, F. (1992) Aproximacio´n aleatoria de cuerpos convexos. Publ. Mat. Barc. 36: 85–109[2] BÁRÁNY, I., BUCHTA, C. (1993) Random polytopes in a convex polytope, independence of shape, and concentration of vertices. Math. Ann. 297: 467–497[3] BUCHTA, C. (1985) Zufällige Polyeder – Eine Übersicht. In: HLAWKA, E. (ed.) Zahlentheoretische Analysis, pp. 1–13. Lecture Notes in Mathematics,Vol. 1114, Springer, Berlin[4] BUCHTA, C. (2005) An identity relating moments of functionals of convex hulls. Discrete Comput. Geom. 33: 125–142[5] BUCHTA, C. (2006) The exact distribution of the number of vertices of a random convex chain. Mathematika 53: 247–254[6] BUCHTA, C., REITZNER, M. (1997) Equiaffine inner parallel curves of a plane convex body and the convex hulls of randomly chosen points. Probab. Theory Relat. Fields 108: 385–415[7] BUCHTA, C., REITZNER, M. (2001) The convex hull of random points in a tetrahedron: Solution of Blaschke’s problem and more general results. J. Reine Angew. Math. 536: 1–29[8] GRUBER, P.M. (1997) Comparisons of best and random approximation of convex bodies by polytopes. Rend. Circ. Mat. Palermo (2) Suppl. 50: 189–216[9] MATHAI,A.M. (1999) An Introduction to Geometrical Probability. Distributional Aspects with Applications. Gordon and Breach, Amsterdam