Random points on the boundary of smooth convex bodies

Reitzner, Matthias

doi:10.1090/S0002-9947-02-02962-8

Random points on the boundary of smooth convex bodies
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by Matthias Reitzner

Trans. Amer. Math. Soc. 354 (2002), 2243-2278

DOI: https://doi.org/10.1090/S0002-9947-02-02962-8

Published electronically: February 7, 2002

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Abstract:

The convex hull of $n$ independent random points chosen on the boundary of a convex body $K \subset \mathbb {R}^d$ according to a given density function is a random polytope. The expectation of its $i$–th intrinsic volume for $i=1, \dots , d$ is investigated. In the case that the boundary of $K$ is sufficiently smooth, asymptotic expansions for these expected intrinsic volumes as $n \to \infty$ are derived.

References

F. Affentranger, The convex hull of random points with spherically symmetric distributions, Rend. Sem. Mat. Univ. Politec. Torino 49 (1991), no. 3, 359–383 (1993). MR 1231058
Fernando Affentranger, Random approximation of convex bodies, Publ. Mat. 36 (1992), no. 1, 85–109 (Spanish, with English summary). MR 1179605, DOI 10.5565/PUBLMAT_{3}6192_{0}8
Imre Bárány, Random polytopes in smooth convex bodies, Mathematika 39 (1992), no. 1, 81–92. MR 1176473, DOI 10.1112/S0025579300006872
Imre Bárány and Christian Buchta, Random polytopes in a convex polytope, independence of shape, and concentration of vertices, Math. Ann. 297 (1993), no. 3, 467–497. MR 1245400, DOI 10.1007/BF01459511
Christian Buchta, Zufällige Polyeder—eine Übersicht, Zahlentheoretische Analysis, Lecture Notes in Math., vol. 1114, Springer, Berlin, 1985, pp. 1–13 (German). MR 1031808, DOI 10.1007/BFb0101638
C. Buchta, J. Müller, and R. F. Tichy, Stochastical approximation of convex bodies, Math. Ann. 271 (1985), no. 2, 225–235. MR 783553, DOI 10.1007/BF01455988
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
Peter M. Gruber, Expectation of random polytopes, Manuscripta Math. 91 (1996), no. 3, 393–419. MR 1416720, DOI 10.1007/BF02567963
Peter M. Gruber, Comparisons of best and random approximation of convex bodies by polytopes, Rend. Circ. Mat. Palermo (2) Suppl. 50 (1997), 189–216. II International Conference in “Stochastic Geometry, Convex Bodies and Empirical Measures” (Agrigento, 1996). MR 1602970
Peter Henrici, Applied and computational complex analysis, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Volume 1: Power series—integration—conformal mapping—location of zeros. MR 0372162
Daniel Hug, Absolute continuity for curvature measures of convex sets. II, Math. Z. 232 (1999), no. 3, 437–485. MR 1719698, DOI 10.1007/PL00004765
R. E. Miles, Isotropic random simplices, Advances in Appl. Probability 3 (1971), 353–382. MR 309164, DOI 10.2307/1426176
Josef S. Müller, On the mean width of random polytopes, Probab. Theory Related Fields 82 (1989), no. 1, 33–37. MR 997430, DOI 10.1007/BF00340011
Josef S. Müller, Approximation of a ball by random polytopes, J. Approx. Theory 63 (1990), no. 2, 198–209. MR 1079850, DOI 10.1016/0021-9045(90)90103-W
Reitzner, M.: Stochastical approximation of smooth convex bodies. Mathematika, to appear.
A. Rényi and R. Sulanke, Über die konvexe Hülle von $n$ zufällig gewählten Punkten, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1963), 75–84 (1963) (German). MR 156262, DOI 10.1007/BF00535300
A. Rényi and R. Sulanke, Über die konvexe Hülle von $n$ zufällig gewählten Punkten. II, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 3 (1964), 138–147 (1964) (German). MR 169139, DOI 10.1007/BF00535973
E. Harzheim, A combinatorial theorem which is related to the invariance of the separating set for the plane, Topics in combinatorics and graph theory (Oberwolfach, 1990) Physica, Heidelberg, 1990, pp. 347–352. MR 1100054
Luis A. Santaló, Integral geometry and geometric probability, Encyclopedia of Mathematics and its Applications, Vol. 1, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. With a foreword by Mark Kac. MR 0433364
Rolf Schneider, Zur optimalen Approximation konvexer Hyperflächen durch Polyeder, Math. Ann. 256 (1981), no. 3, 289–301 (German). MR 626950, DOI 10.1007/BF01679698
Schneider, R.: Random approximation of convex sets. J. Microscopy 151, 211–227 (1988).
Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521, DOI 10.1017/CBO9780511526282
Jacob E. Goodman and Joseph O’Rourke (eds.), Handbook of discrete and computational geometry, CRC Press Series on Discrete Mathematics and its Applications, CRC Press, Boca Raton, FL, 1997. MR 1730156
Carsten Schütt, Random polytopes and affine surface area, Math. Nachr. 170 (1994), 227–249. MR 1302377, DOI 10.1002/mana.19941700117
Carsten Schütt and Elisabeth Werner, Random polytopes with vertices on the boundary of a convex body, C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), no. 9, 697–701 (English, with English and French summaries). MR 1797754, DOI 10.1016/S0764-4442(00)01685-2
Wolfgang Weil and John A. Wieacker, Stochastic geometry, Handbook of convex geometry, Vol. A, B, North-Holland, Amsterdam, 1993, pp. 1391–1438. MR 1243013
Wieacker, J. A.: Einige Probleme der polyedrischen Approximation. Diplomarbeit, Freiburg im Breisgau 1978.
M. Zähle, A kinematic formula and moment measures of random sets, Math. Nachr. 149 (1990), 325–340. MR 1124814, DOI 10.1002/mana.19901490125