Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T21:04:42.608Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastical approximation of smooth convex bodies

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Matthias Reitzner
Matthias Reitzner
Affiliation:
Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstrasse 8/104, A-1040 Wien, Austria
Get access

Abstract

A random polytope is the convex hull of n random points in the interior of a convex body K. The expectation of the ith intrinsic volume of a random polytope as n → ∞ is investigated. It is proved that, for convex bodies of differentiability class Kk+1, precise asymptotic expansions for these expectations exist. The proof makes essential use of a refinement of Crofton's boundary theorem.

MSC classification

Secondary: 52A22: Random convex sets and integral geometry
Type
Research Article
Information
Mathematika , Volume 51 , Issue 1-2 , December 2004 , pp. 11 - 29
Copyright
Copyright © University College London 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Affentranger, F.. The expected volume of a random polytope in a ball. J. Microscopy. 151 (1988), 277–287.CrossRefGoogle Scholar
2.Affentranger, F.. Aproximacion aleatoria de cuerpos convexos. Publ. Mat. Bare. 36 (1992). 85–109.CrossRefGoogle Scholar
3.Alikoski, H. A.. Über das Sylvestersche Vierpunktproblem. Ann. Accul. Sci. Fennicac. 51 (1939. 1–10.Google Scholar
4.Baddeley, A.. Integrals on a moving manifold and geometric probability. Adv. Appl. Prob. 9 (1977), 588–603.CrossRefGoogle Scholar
5.Bàràny, I.. Random polytopes in smooth convex bodies. Mathematika, 39 (1992), 81–92.CrossRefGoogle Scholar
6.Bàràny, I. and Buchta, C.. Random polytopes in a convex polytope, independence of shape. and concentration of vertices. Math. Ann., 297 (1993), 467–497.CrossRefGoogle Scholar
7.Buchta, C.. Stochastische Approximation konvexer Polygone. Z. Wahrscheiidichkeitsth. lent. Geb., 67 (1984), 283–304.CrossRefGoogle Scholar
8.Buchta, C.. Zufallspolygone in konvexen Vielecken. J reine angew. Math. 347 (1984). 212–220.Google Scholar
9.Buchta, C.. Das Volumen von Zufallspolyedern im Ellipsoid. AnZ. Österr. Akad. Wiss. Math.- Naturwiss. Kl., 121 (1984), 1–4.Google Scholar
10.Buchta, C. and Muller, J.. Random polytopes in a ball, J. Appl. Prohab. 21 (1984). 753–762.CrossRefGoogle Scholar
11.Buchta, C. and Reitzner, M.. What is the expected volume of a tetrahedron whose vertices are chosen at random from a given tetrahedron? An:. Österr. Akad. Wiss.. Math.-Natunviss. KL., 129 (1992), 63–68.Google Scholar
12.Buchta, C. and Reitzner, M.. Equiaffine inner parallel curves of a plane convex body and the convex hulls of randomly chosen points. Probab. Theory Relat. Fields, 108 (1997). 385–415.CrossRefGoogle Scholar
13.Buchta, C. and Reitzner, M.. The convex hull of random points in a tetrahedron: solution of Blaschke's problem and more general results. J. reine angew. Math.. 536 (2001), 129.CrossRefGoogle Scholar
14.Crofton, M. W.. Probability. Encyclopaedia Britannica, 9th ed, vol. 19 (1885); 768–788.Google Scholar
15.Dcltheil, R.. Probabilités géométriques. In Traité du Calcul des Probabilites et de ses Applications, vol. II, fasc II (ed. Borel, E.). Gauthiers-Villars (Paris, 1926).Google Scholar
16.Doetsch, G.. Handbuch der Laplace-Transformation, II. Birkhäuser (Basel-Stuttgart. 1955).CrossRefGoogle Scholar
17.Efron, B.. The convex hull of a random set of points. Biometrika., 52 (1965), 331–343.CrossRefGoogle Scholar
18.Gruber, P. M.. Expectation of random polytopes. Manuscripta Math., 91 (1996). 393–419.CrossRefGoogle Scholar
19.Gruber, P. M.. Comparisons of best and random approximation of convex bodies by polytopes. Rend. Circ. Mat. Palermo, II. Ser., Suppl., 50 (1997), 189–216.Google Scholar
20.Henrici, P.. Applied and Computational Complex Analysis, Pure and Applied Mathematics. Wiley-Interscience (New York, 1974).Google Scholar
21.Hostinsky, B.. Sur les Probabilités Géométriques. Publ. Fac. Sci. Univ. Masaryk (Brno, 1925).Google Scholar
22.Hug, D.. Absolute continuity for curvature measures of convex sets, II. Math. Z.., 232 (1999). 437–485.CrossRefGoogle Scholar
23.Kendall, M. G. and Moran, P. A. P.. Geometrical Probability, Griffin (London. 1963).Google Scholar
24.Kingman, J. F. C.. Random secants of a convex body. J. Appl. Probab., 6 (1969), 660–672.CrossRefGoogle Scholar
25.Mannion, D.. The volume of a tetrahedron whose vertices are chosen at random in the interior of a parent tetrahedron. Adv. Appl. Probab., 26 (1994), 577596.CrossRefGoogle Scholar
26.Reed, W. J. and Ruben, H.. A more general form of a theorem of Crofton. J. Appl. Probab.. 10 (1973), 479–482.Google Scholar
27.Reitzner, M.. The floating body and the equiaffine inner parallel curve of a plane convex body. Geom. Dedicata, 84 (2001), 151–167.CrossRefGoogle Scholar
28.Rényi, A. and Sulanke, R.. Über die konvexe HÜlle von n zufällig gewählten Punkten. II. Z. Wahrscheinlichkeitsth. Verw. Geb., 2 (1963), 75–84.CrossRefGoogle Scholar
29.Renyi, A. and Sulanke, R.. Über die konvexe Hülle von n zufällig gewahlten Punkten II. Z. Wahrscheinlichkeitsth. Verw. Geb, 3 (1964), 138–147.CrossRefGoogle Scholar
30.Santalo, L. A.. Integral Geometry and Geometric Probability. Addison-Wesley (Reading MA, 1976).Google Scholar
31.Schneider, R.. Zur optimalc Approximation konvexer Hyperflächen durch Polyeder. Math. Ann., 256 (1981) 289–301.CrossRefGoogle Scholar
32.Schneider, R.. Approximation of convex bodies by random polytopes. Aequationes Math., 32 (1987), 304–310.CrossRefGoogle Scholar
33.Schneider, R.. Convex Bodies: The Brunn-Minkowski theory. Encyclopedia of Mathematics and Its Applications 44 Cambridge University Press (Cambridge, 1993).CrossRefGoogle Scholar
34.Schneider, R.. Discrete aspects of stochastic geometry. In: Goodman, O'Rourke, J. E., J. (eds.) Handbook of Discrete and Computational Geometry, pp. 167–184. Boca Raton: CRC Press 1997 (CRC Press Series on Discrete Mathematics and its Applications).Google Scholar
35.Schneider, R. and Wieacker, J. A.. Random polytopes in a convex body. Z. Wahrscheinlichkeitsth, Verw. Geb., 52 (1980), 69–73.CrossRefGoogle Scholar
36.Schiitt, C.. Random polytopes and affine surface area. Math. Nachr., 170 (1994), 227–249.CrossRefGoogle Scholar
37.Solomon, H.. Geometric Probability. (Regional Conference Series in Applied Mathematics 28), SI AM (Philadelphia PA, 1989).Google Scholar
38.Weil, W. and Wieacker, J. A.. Stochastic geometry. In Handbook of Convex Geometry (ed. Gruber, P. M. and Wills, J. M.). North-Holland/Elsevier (Amsterdam-London-New York-Tokyo, 1993), 1391–1438.CrossRefGoogle Scholar
39.Wieacker, J. A.. Einige Probleme der Polyedrischen Approximation. Diplomarbeit, Albert- Ludwigs-Universität Freiburg (1978).Google Scholar
40.Woolhouse, W.. Educational Times (1867).Google Scholar