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Affentranger, F.: The expected volume of a random polytope in a ball. J. Microscopy151, 277–287 (1988)
Affentranger, F.: Aproximación aleatoria de cuerpos convexos. Publ. Mat., Barc.36, 85–109 (1992)
Affentranger, F., Wieacker, J. A.: On the convex hull of uniform random points in a simpled-polytope. Discrete Comput. Geom.6, 291–305 (1991)
Bárány, I.: Intrinsic volumes andf-vectors of random polytopes. Math. Ann.285, 671–699 (1989)
Bárány, I., Buchta, C.: On the convex hull of uniform random points in an arbitraryd-polytope. Anz. Österr. Akad. Wiss., Math.-Naturwiss. Kl.127, 25–27 (1990)
Bárány, I., Larman, D. G.: Convex bodies, economic cap coverings, random polytopes. Mathematika35, 274–291 (1988)
Bayer, M. M., Lee, C. W.: Convex polytopes. In: Gruber, P. M., Wills, J. M. (eds.) Handbook of convex geometry. Amsterdam: Elsevier 1993
Bentley, J. L., Kung, H. T., Schkolnick, M., Thompson, C. D.: On the average number of maxima in a set of vectors and applications. J. Assoc. Comput. Math.25, 536–543 (1978)
Blaschke, W.: Über affine Geometrie XI: Lösung des “Vierpunktproblems” von Sylvester aus der Theorie der geometrischen Wahrscheinlichkeiten. Ber. Verh. Sächs. Ges. Wiss. Leipzig, Math.-Phys. Kl.69, 436–453 (1917)
Blaschke, W.: Vorlesungen über Differentialgeometrie, vol. II. Affine Differentialgeometrie. Berlin: Springer 1923
Buchta, C.: Über die konvexe Hülle von Zufallspunkten in Eibereichen. Elem. Math.38, 153–156 (1983)
Buchta, C.: Zufallspolygone in konvexen Vielecken. J. Reine Angew. Math.347, 212–220 (1984)
Buchta, C.: Zufallige Polyeder-Eine Übersicht. In: Hlawka, E. (ed.) Zahlentheoretische Analysis (Lect. Notes Math., vol. 1114, pp. 1–13) Berlin Heidelberg New York Tokyo: Springer 1985
Buchta, C.: A note on the volume of a random polytope in a tetrahedron. Illinois J. Math.30, 653–659 (1986)
Buchta, C., Reitzner, M.: What is the expected volume of a tetrahedron whose vertices are chosen at random from a given tetrahedron? Anz. Österr. Akad. Wiss., Math.-Naturwiss. Kl.129, 63–68 (1992)
Croft, H. T., Falconer, K. J., Guy, R. K.: Unsolved problems in geometry. New York Berlin Heidelberg: Springer 1991
Dalla, L., Larman, D. G.: Volumes of a random polytope in a convex set. In: Gritzmann, P., Sturmfels, B. (eds.) Applied geometry and discrete mathematics: The Victor Klee Festschrift (DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 4, pp. 175–180) Providence, RI: Am. Math. Soc. 1991
Devroye, L.: A note on finding convex hulls via maximal vectors. Inf. Process. Lett.11, 53–56 (1980)
Dwyer, R. A.: On the convex hull of random points in a polytope. J. Appl. Probab.25, 688–699 (1988)
Dwyer, R. A., Kannan, R.: Convex hull of randomly chosen points from a polytope. Math. Res.38, 16–24 (1987)
Efron, B.: The convex hull of a random set of points. Biometrika52, 331–343 (1965)
Ewald, G., Larman, D. G., Rogers, C. A.: The directions of the line segments and of the υ-dimensional balls on the boundary of a convex body in Euclidean space. Mathematika17, 1–20 (1970)
Feller, W.: An introduction to probability theory and its applications, vol. II. New York London Sydney: Wiley 1966
Giannopoulos, A. A.: On the mean value of the area of a random polygon in a plane convex body. Mathematika39, 279–290 (1992)
Groemer, H.: On the mean value of the volume of a random polytope in a convex set. Arch. Math.25, 86–90 (1974)
Groemer, H.: Math. Reviews 84g: 60019 (1984)
Macbeath, A. M.: A theorem on non-homogeneous lattices. Ann. Math., II. Ser.56, 269–293 (1952)
Pfiefer, R. E.: The historical development of J. J. Sylvester's four point problem. Math. Mag.62, 309–317 (1989)
Reed, W. J.: Random points in a simplex. Pacific J. Math.54, 183–198 (1974)
Rényi, A., Sulanke, R.: Über die konvexe Hülle vonn zufällig gewählten Punkten. Z. Wahrscheinlichkeitsth. Verw. Geb.2, 75–84 (1963)
Schneider, R.: Random approximation of convex sets. J. Microscopy151, 211–227 (1988)
Schütt, C.: The convex floating body and polyhedral approximation. Israel. J. Math.73, 65–77 (1991)
Van Wel, B. F.: The convex hull of a uniform sample from the interior of a simpled-polytope. J. Appl. Probab.26, 259–273 (1989)
Weil, W., Wieacker, J. A.: Stochastic geometry. In: Gruber, P. M., Wills, J. M. (eds.) Handbook of convex geometry. Amsterdam: Elsevier 1993
Wieacker, J. A.: Einige Probleme der polyedrischen Approximation. Diplomarbeit, Albert-Ludwigs-Universität, Freiburg im Breisgau 1978
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Supported by the Program in Discrete Mathematics and Its Applications at Yale and NSF Grant CCR-8901484. Partially supported by the Hungarian National Science Foundation Grant 1812
Work on this paper was done during a stay at the University of Freiburg im Breisgau supported by Deutscher Akademischer Austauschdienst
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Bárány, I., Buchta, C. Random polytopes in a convex polytope, independence of shape, and concentration of vertices. Math. Ann. 297, 467–497 (1993). https://doi.org/10.1007/BF01459511
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DOI: https://doi.org/10.1007/BF01459511