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Stochastische Approximation konvexer Polygone. (German) Zbl 0536.60020
Let k be a convex polygon with perimeter L and area F. Denote the perimeter of the convex hull of n uniformly and independently distributed points in K by \(L_ n\) and by \(F_ n\) its area. As n tends to infinity, \(L-E(L_ n)=C_ 1(K)/\sqrt{n}+o(1/n^{1-\epsilon})\) for any fixed \(\epsilon>0\) and \(F-E(F_ n)=C_ 2(K)\log n/n+0(1/n)\), where the constants \(C_ 1(K)\) and \(C_ 2(K)\) are given explicitly. This generalizes a result of A. Rényi and R. Sulanke, ibid. 2, 75-84 (1963; Zbl 0118.137) and ibid. 3, 138-147 (1964; Zbl 0126.341).

MSC:
60D05 Geometric probability and stochastic geometry
52A22 Random convex sets and integral geometry (aspects of convex geometry)
60F99 Limit theorems in probability theory
Keywords:
convex polygon
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[1] Baddeley, A.: A fourth note on recent research in geometrical probability. Adv. Appl. Probability 9, 824-860 (1977) · Zbl 0387.60019 · doi:10.2307/1426702
[2] Buchta, C.: Das Volumen von Zufallspolyedern im Ellipsoid. Anz. Österr. Akad. Wiss. Math. ? Natur. Kl. 1984 · Zbl 0574.52006
[3] Czuber, E.: Wahrscheinlichkeitsrechnung und ihre Anwendung auf Fehlerausgleichung, Statistik und Lebensversicherung I. Leipzig-Berlin: Teubner 1903 · JFM 33.0236.01
[4] Efron, B.: The convex hull of a random set of points. Biometrika 52, 331-343 (1965) · Zbl 0138.41301
[5] Groemer, H.: On some mean values associated with a randomly selected simplex in a convex set. Pacific J. Math. 45, 525-533 (1973) · Zbl 0258.52004
[6] Groemer, H.: On the mean value of the volume of a random polytope in a convex set. Arch. Math. 25, 86-90 (1974) · Zbl 0287.52009 · doi:10.1007/BF01238645
[7] Gruber, P.M.: Approximation of convex bodies. In: Gruber, P.M., Wills, J.M., eds.: Convexity and its applications. Basel: Birkhäuser 1983 · Zbl 0519.52005
[8] Kendall, M.G., Moran, P.A.P.: Geometrical probability. London: Griffin 1963 · Zbl 0105.35002
[9] Kingman, J.F.C.: Random secants of a convex body. J. Appl. Probability 6, 660-672 (1969) · Zbl 0186.51603 · doi:10.2307/3212110
[10] Klee, V.: What is the expected volume of a simplex whose vertices are chosen at random from a given convex body? Amer. Math. Monthly 76, 286-288 (1969) · doi:10.2307/2316377
[11] Little, D.V.: A third note on recent research in geometrical probability. Adv. Appl. Probability 6, 103-130 (1974) · Zbl 0293.60015 · doi:10.2307/1426209
[12] Moran, P.A.P.: A note on recent research in geometrical probability. J. Appl. Probability 3, 453-463 (1966) · Zbl 0168.17008 · doi:10.2307/3212131
[13] Moran, P.A.P.: A second note on recent research in geometrical probability. Adv. Appl. Probability 1, 73-89 (1969) · Zbl 0181.45104 · doi:10.2307/1426409
[14] Rényi, A., Sulanke, R.: Über die konvexe Hülle von n zufällig gewählten Punkten. Z. Wahrscheinlichkeitstheorie verw. Gebiete 2, 75-84 (1963) · Zbl 0118.13701 · doi:10.1007/BF00535300
[15] Rényi, A., Sulanke, R.: Über die konvexe Hülle von n zufällig gewählten Punkten II. Z. Wahrscheinlichkeitstheorie verw. Gebiete 3, 138-147 (1964) · Zbl 0126.34103 · doi:10.1007/BF00535973
[16] Ripley, B.D., Rasson, J.-P.: Finding the edge of a poisson forest. J. Appl. Probability 14, 483-491 (1977) · Zbl 0373.62058 · doi:10.2307/3213451
[17] Santaló, L.A.: Integral geometry and geometric probability. Reading, Massachusetts: Addison-Wesley 1976 · Zbl 0342.53049
[18] Schneider, R., Wieacker, J.A.: Random polytopes in a convex body. Z. Wahrscheinlichkeitstheorie verw. Gebiete 52, 69-73 (1980) · Zbl 0407.60009 · doi:10.1007/BF00534188
[19] Schöpf, P.: Gewichtete Volumsmittelwerte von Simplices, welche zufällig in einem konvexen Körper des ?n gewählt werden. Monatsh. Math. 83, 331-337 (1977) · Zbl 0383.52011 · doi:10.1007/BF01387909
[20] Wieacker, J.A.: Einige Probleme der polyedrischen Approximation. Freiburg im Breisgau: Diplomarbeit 1978
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