메뉴 건너뛰기




Volumn , Issue , 2009, Pages

Random Polytopes

Author keywords

Convex hulls; Deviation; Functionals; Limit theorems; Random points; Random polytopes

Indexed keywords


EID: 84920034477     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1093/acprof:oso/9780199232574.003.0002     Document Type: Chapter
Times cited : (8)

References (93)
  • 1
    • 0010041993 scopus 로고
    • The convex hull of random points with spherically symmetric distributions
    • Affentranger, F. (1991) The convex hull of random points with spherically symmetric distributions. Rend. Sem. Mat. Univ. Politec. Torino, 49, 359-383.
    • (1991) Rend. Sem. Mat. Univ. Politec. Torino , vol.49 , pp. 359-383
    • Affentranger, F.1
  • 3
    • 0040023867 scopus 로고
    • Intrinsic volumes and f-vectors of random polytopes
    • Bárány, I. (1989) Intrinsic volumes and f-vectors of random polytopes. Math. Ann., 285, 671-699.
    • (1989) Math. Ann. , vol.285 , pp. 671-699
    • Bárány, I.1
  • 4
    • 84976132344 scopus 로고
    • Random polytopes in smooth convex bodies
    • Bárány, I. (1992) Random polytopes in smooth convex bodies. Mathematika, 39, 81-92.
    • (1992) Mathematika , vol.39 , pp. 81-92
    • Bárány, I.1
  • 5
    • 85052071122 scopus 로고    scopus 로고
    • Corrigendum: Random polytopes in smooth convex bodies
    • Bárány, I. (2004) Corrigendum: Random polytopes in smooth convex bodies. Mathematika, 51, 31.
    • (2004) Mathematika , vol.51 , pp. 31
    • Bárány, I.1
  • 6
    • 0000683778 scopus 로고
    • Random polytopes in a convex polytope, independence of shape, and concentration of vertices
    • Bárány, I. and Buchta, C. (1993) Random polytopes in a convex polytope, independence of shape, and concentration of vertices. Math. Ann., 297, 467-497.
    • (1993) Math. Ann. , vol.297 , pp. 467-497
    • Bárány, I.1    Buchta, C.2
  • 7
    • 84971809021 scopus 로고
    • Convex bodies, economic cap coverings, random polytopes
    • Bárány, I. and Larman, D. G. (1988) Convex bodies, economic cap coverings, random polytopes. Mathematika, 35, 274-291.
    • (1988) Mathematika , vol.35 , pp. 274-291
    • Bárány, I.1    Larman, D.G.2
  • 8
    • 0035839380 scopus 로고    scopus 로고
    • On 0-1 polytopes with many facets
    • Bárány, I. and Pór, A. (2001) On 0-1 polytopes with many facets. Adv. Math., 161, 209-228.
    • (2001) Adv. Math. , vol.161 , pp. 209-228
    • Bárány, I.1    Pór, A.2
  • 10
    • 41649083417 scopus 로고    scopus 로고
    • Central limit theorems for Gaussian polytopes
    • Bárány, I. and Vu, H. V. (2007) Central limit theorems for Gaussian polytopes. Ann. Probab., 35, 1593-1621.
    • (2007) Ann. Probab. , vol.35 , pp. 1593-1621
    • Bárány, I.1    Vu, H.V.2
  • 12
    • 0347348614 scopus 로고    scopus 로고
    • On the Hausdorff distance between a convex set and an interior random convex hull
    • Bingham, N. H., Bräker, H., and Hsing, T. (1998) On the Hausdorff distance between a convex set and an interior random convex hull. Adv. Appl. Prob., 30, 295-316.
    • (1998) Adv. Appl. Prob. , vol.30 , pp. 295-316
    • Bingham, N.H.1    Bräker, H.2    Hsing, T.3
  • 13
    • 0037946210 scopus 로고
    • Über affine Geometrie XI: Lösung des "Vierpunktproblems" von Sylvester aus der Theorie der geometrischen Wahrscheinlichkeiten
    • Reprinted in: Burau, W., et al. (eds.): Wilhelm Blaschke. Gesammelte Werke, vol. 3: Konvexgeometrie. pp. 284-301. Essen, Thales 1985
    • Blaschke, W. (1917) 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. Reprinted in: Burau, W., et al. (eds.): Wilhelm Blaschke. Gesammelte Werke, vol. 3: Konvexgeometrie. pp. 284-301. Essen, Thales 1985.
    • (1917) Ber. Verh. Sächs. Ges. Wiss. Leipzig, Math.-Phys. Kl. , vol.69 , pp. 436-453
    • Blaschke, W.1
  • 15
    • 0033273787 scopus 로고    scopus 로고
    • Random projections of regular polytopes
    • Böröczky Jr., K. and Henk, M. (1999) Random projections of regular polytopes. Arch. Math., 73, 465-473.
    • (1999) Arch. Math. , vol.73 , pp. 465-473
    • Böröczky, K.1    Henk, M.2
  • 16
    • 21144457532 scopus 로고    scopus 로고
    • Approximation of smooth convex bodies by random circumscribed polytopes
    • Böröczky Jr., K. and Reitzner, M. (2004) Approximation of smooth convex bodies by random circumscribed polytopes. Ann. Appl. Probab., 14, 239-273.
    • (2004) Ann. Appl. Probab. , vol.14 , pp. 239-273
    • Böröczky, K.1    Reitzner, M.2
  • 18
    • 0041063389 scopus 로고
    • Stochastische Approximation konvexer Polygone
    • Buchta, C. (1984) Stochastische Approximation konvexer Polygone. Z. Wahrsch. Verw. Geb., 67, 283-304.
    • (1984) Z. Wahrsch. Verw. Geb. , vol.67 , pp. 283-304
    • Buchta, C.1
  • 19
    • 84920030917 scopus 로고
    • On the number of vertices of random polyhedra with a given number of facets
    • Buchta, C. (1987) On the number of vertices of random polyhedra with a given number of facets. SIAM J. Alg. Disc. Meth., 8, 85-92.
    • (1987) SIAM J. Alg. Disc. Meth. , vol.8 , pp. 85-92
    • Buchta, C.1
  • 20
    • 11944265105 scopus 로고    scopus 로고
    • An identity relating moments of functionals of convex hulls
    • Buchta, C. (2005) An identity relating moments of functionals of convex hulls. Discrete Comput. Geom., 33, 125-142.
    • (2005) Discrete Comput. Geom. , vol.33 , pp. 125-142
    • Buchta, C.1
  • 21
    • 0000234482 scopus 로고
    • Stochastical approximation of convex bodies
    • Buchta, C., Müller, J., and Tichy, R. F. (1985) Stochastical approximation of convex bodies. Math. Ann., 271, 225-235.
    • (1985) Math. Ann. , vol.271 , pp. 225-235
    • Buchta, C.1    Müller, J.2    Tichy, R.F.3
  • 22
    • 21844507773 scopus 로고
    • Limit theorems for functionals of convex hulls
    • Cabo, A. J. and Groeneboom, P. (1994) Limit theorems for functionals of convex hulls. Probab. Theory Relat. Fields, 100, 31-55.
    • (1994) Probab. Theory Relat. Fields , vol.100 , pp. 31-55
    • Cabo, A.J.1    Groeneboom, P.2
  • 24
    • 29144439194 scopus 로고    scopus 로고
    • Decoding by linear programming
    • Candes, E. and Tao, T. (2005) Decoding by linear programming. IEEE Trans. Inform. Theory, 51, 4203-4215.
    • (2005) IEEE Trans. Inform. Theory , vol.51 , pp. 4203-4215
    • Candes, E.1    Tao, T.2
  • 26
    • 35948969765 scopus 로고    scopus 로고
    • Identities linking volumes of convex hulls
    • Cowan, R. (2007) Identities linking volumes of convex hulls. Adv. Appl. Prob., 39, 630-644.
    • (2007) Adv. Appl. Prob. , vol.39 , pp. 630-644
    • Cowan, R.1
  • 27
  • 28
    • 0042343178 scopus 로고
    • Volumes of a random polytope in a convex set. In: Gritzmann, P., Sturmfels, B. (eds.): Applied geometry and discrete mathematics. The Victor Klee Festschrift
    • Dalla, L. and Larman, D. G. (1991) 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, 4, 175-180.
    • (1991) DIMACS Series in Discrete Mathematics and Theoretical Computer Science , vol.4 , pp. 175-180
    • Dalla, L.1    Larman, D.G.2
  • 29
    • 33646479497 scopus 로고    scopus 로고
    • High-dimensional centrally symmetric polytopes with neighbourliness proportional to dimension
    • Donoho, D. L. (2006) High-dimensional centrally symmetric polytopes with neighbourliness proportional to dimension. Discrete Comput. Geom., 35, 617-652.
    • (2006) Discrete Comput. Geom. , vol.35 , pp. 617-652
    • Donoho, D.L.1
  • 30
    • 22144471532 scopus 로고    scopus 로고
    • Neighborliness of randomly projected simplices in high dimensions
    • Donoho, D. L. and Tanner, J. (2005a). Neighborliness of randomly projected simplices in high dimensions. Proc. Natl. Acad. Sci. USA, 102, 9452-9457.
    • (2005) Proc. Natl. Acad. Sci. USA , vol.102 , pp. 9452-9457
    • Donoho, D.L.1    Tanner, J.2
  • 31
    • 22144488449 scopus 로고    scopus 로고
    • Sparse nonnegative solution of underdetermined linear equations by linear programming
    • Donoho, D. L. and Tanner, J. (2005b). Sparse nonnegative solution of underdetermined linear equations by linear programming. Proc. Natl. Acad. Sci. USA, 102(27), 9446-9451.
    • (2005) Proc. Natl. Acad. Sci. USA , vol.102 , Issue.27 , pp. 9446-9451
    • Donoho, D.L.1    Tanner, J.2
  • 32
    • 57349138185 scopus 로고    scopus 로고
    • Counting faces of randomly projected polytopes when the projection radically lowers dimension
    • Donoho, D. L. and Tanner, J. (2009) Counting faces of randomly projected polytopes when the projection radically lowers dimension. J. Amer. Math. Soc., 22, 1-53.
    • (2009) J. Amer. Math. Soc. , vol.22 , pp. 1-53
    • Donoho, D.L.1    Tanner, J.2
  • 33
    • 0008364462 scopus 로고    scopus 로고
    • Rates of convergence for random approximations of convex sets
    • Dümbgen, L. and Walther, G. (1996) Rates of convergence for random approximations of convex sets. Adv. Appl. Prob., 28, 981-986.
    • (1996) Adv. Appl. Prob. , vol.28 , pp. 981-986
    • Dümbgen, L.1    Walther, G.2
  • 35
    • 0003516674 scopus 로고
    • EATCS Monographs on Theoretical Computer Science, Springer, Berlin
    • Edelsbrunner, H. (1987). Algorithms in Combinatorial Geometry, Volume 10 of EATCS Monographs on Theoretical Computer Science. Springer, Berlin.
    • (1987) Algorithms in Combinatorial Geometry , vol.10
    • Edelsbrunner, H.1
  • 36
    • 0000788596 scopus 로고
    • The convex hull of a random set of points
    • Efron, B. (1965) The convex hull of a random set of points. Biometrika, 52, 331-343.
    • (1965) Biometrika , vol.52 , pp. 331-343
    • Efron, B.1
  • 37
    • 13344285317 scopus 로고    scopus 로고
    • Random convex hulls: a variance revisited
    • Finch, S. and Hueter, I. (2004) Random convex hulls: a variance revisited. Adv. Appl. Prob., 36, 981-986.
    • (2004) Adv. Appl. Prob. , vol.36 , pp. 981-986
    • Finch, S.1    Hueter, I.2
  • 38
    • 23944490670 scopus 로고    scopus 로고
    • Lower bound for the maximal number of facets of a 0/1 polytope
    • Gatzouras, D., Giannopoulos, A., and Markoulakis, N. (2005) Lower bound for the maximal number of facets of a 0/1 polytope. Discrete Comput. Geom., 34, 331-349.
    • (2005) Discrete Comput. Geom. , vol.34 , pp. 331-349
    • Gatzouras, D.1    Giannopoulos, A.2    Markoulakis, N.3
  • 40
    • 0000664336 scopus 로고
    • Localisation asymptotique du polyèdre d'appui d'un déchantillon Laplacien dà k dimensions
    • Geffroy, J. (1961) Localisation asymptotique du polyèdre d'appui d'un déchantillon Laplacien dà k dimensions. Publ. Inst. Statist. Univ. Paris, 10, 213-228.
    • (1961) Publ. Inst. Statist. Univ. Paris , vol.10 , pp. 213-228
    • Geffroy, J.1
  • 41
    • 84971985346 scopus 로고
    • On the mean value of the area of a random polygon in a plane convex body
    • Giannopoulos, A. (1992) On the mean value of the area of a random polygon in a plane convex body. Mathematika, 39, 279-290.
    • (1992) Mathematika , vol.39 , pp. 279-290
    • Giannopoulos, A.1
  • 42
    • 0036025567 scopus 로고    scopus 로고
    • Random spaces generated by vertices of the cube
    • Giannopoulos, A. and Hartzoulaki, M. (2002) Random spaces generated by vertices of the cube. Discrete Comput. Geom., 28, 255-273.
    • (2002) Discrete Comput. Geom. , vol.28 , pp. 255-273
    • Giannopoulos, A.1    Hartzoulaki, M.2
  • 43
    • 84972556177 scopus 로고
    • On some mean values associated with a randomly selected simplex in a convex set
    • Groemer, H. (1973) On some mean values associated with a randomly selected simplex in a convex set. Pacific J. Math., 45, 525-533.
    • (1973) Pacific J. Math. , vol.45 , pp. 525-533
    • Groemer, H.1
  • 44
    • 0013465628 scopus 로고
    • On the mean value of the volume of a random polytope in a convex set
    • Groemer, H. (1974) On the mean value of the volume of a random polytope in a convex set. Arch. Math., 25, 86-90.
    • (1974) Arch. Math. , vol.25 , pp. 86-90
    • Groemer, H.1
  • 45
  • 46
    • 0030292812 scopus 로고    scopus 로고
    • Expectation of random polytopes
    • Gruber, P. M. (1996) Expectation of random polytopes. Manuscripta Math., 91, 393-419.
    • (1996) Manuscripta Math. , vol.91 , pp. 393-419
    • Gruber, P.M.1
  • 47
    • 0038576957 scopus 로고    scopus 로고
    • Quermassintegrals of a random polytope in a convex body
    • Hartzoulaki, M. and Paouris, G. (2003) Quermassintegrals of a random polytope in a convex body. Arch. Math., 80, 430-438.
    • (2003) Arch. Math. , vol.80 , pp. 430-438
    • Hartzoulaki, M.1    Paouris, G.2
  • 48
    • 0001673399 scopus 로고
    • On the asymptotic distribution of the area outside a random convex hull in a disk
    • Hsing, T. (1994) On the asymptotic distribution of the area outside a random convex hull in a disk. Ann. Appl. Probab., 4, 478-493.
    • (1994) Ann. Appl. Probab. , vol.4 , pp. 478-493
    • Hsing, T.1
  • 49
    • 0040153671 scopus 로고
    • The convex hull of a normal sample
    • Hueter, I. (1994) The convex hull of a normal sample. Adv. Appl. Prob., 26, 855-875.
    • (1994) Adv. Appl. Prob. , vol.26 , pp. 855-875
    • Hueter, I.1
  • 50
    • 22844457314 scopus 로고    scopus 로고
    • Limit theorems for the convex hull of random points in higher dimensions
    • Hueter, I. (1999) Limit theorems for the convex hull of random points in higher dimensions. Trans. Amer. Math. Soc., 351, 4337-4363.
    • (1999) Trans. Amer. Math. Soc. , vol.351 , pp. 4337-4363
    • Hueter, I.1
  • 51
    • 8744247425 scopus 로고    scopus 로고
    • Asymptotic mean values of Gaussian polytopes
    • Hug, D., Munsonius, G. O., and Reitzner, M. (2004) Asymptotic mean values of Gaussian polytopes. Beitr. Algebra Geom., 45, 531-548.
    • (2004) Beitr. Algebra Geom. , vol.45 , pp. 531-548
    • Hug, D.1    Munsonius, G.O.2    Reitzner, M.3
  • 52
    • 22244469768 scopus 로고    scopus 로고
    • Gaussian polytopes: variances and limit theorems
    • Hug, D. and Reitzner, M. (2005) Gaussian polytopes: variances and limit theorems. Adv. Appl. Prob., 37, 297-320.
    • (2005) Adv. Appl. Prob. , vol.37 , pp. 297-320
    • Hug, D.1    Reitzner, M.2
  • 54
    • 0017627503 scopus 로고
    • Computer techniques and the archival mapreconstruction of Mycenaean Messenia. In: Bintliff, J. (ed.): Mycenaean geography
    • Kendall, D. (1977) Computer techniques and the archival mapreconstruction of Mycenaean Messenia. In: Bintliff, J. (ed.): Mycenaean geography. Proceedings of the Cambridge Colloquium September 1976, 83-88.
    • (1977) Proceedings of the Cambridge Colloquium September 1976 , pp. 83-88
    • Kendall, D.1
  • 55
    • 65549092263 scopus 로고    scopus 로고
    • On the hyperplane conjecture for random convex sets
    • Klartag, B. and Kozma, G. (2008) On the hyperplane conjecture for random convex sets. Israel J. Math., 170, 253-268.
    • (2008) Israel J. Math. , vol.170 , pp. 253-268
    • Klartag, B.1    Kozma, G.2
  • 56
    • 0001008606 scopus 로고
    • On the approximation of a ball by random polytopes
    • Küfer, K.-H. (1994) On the approximation of a ball by random polytopes. Adv. Appl. Prob., 26, 876-892.
    • (1994) Adv. Appl. Prob. , vol.26 , pp. 876-892
    • Küfer, K.-H.1
  • 57
    • 36549050789 scopus 로고    scopus 로고
    • The limit law of the maximum distance of points in a sphere in R d
    • University of Karlsruhe
    • Lao, W. (2006) The limit law of the maximum distance of points in a sphere in R d . Technical report 2006/10, University of Karlsruhe.
    • (2006) Technical report 2006/10
    • Lao, W.1
  • 60
    • 18644382839 scopus 로고    scopus 로고
    • Smallest singular value of random matrices and geometry of random polytopes
    • Litvak, A. E., Pajor, A., Rudelson, M., and Tomczak-Jaegermann, N. (2005) Smallest singular value of random matrices and geometry of random polytopes. Adv. Math., 195, 491-523.
    • (2005) Adv. Math. , vol.195 , pp. 491-523
    • Litvak, A.E.1    Pajor, A.2    Rudelson, M.3    Tomczak-Jaegermann, N.4
  • 61
    • 0000537779 scopus 로고    scopus 로고
    • A characterization of affine surface area
    • Ludwig, M. and Reitzner, M. (1999) A characterization of affine surface area. Adv. Math., 147, 138-172.
    • (1999) Adv. Math. , vol.147 , pp. 138-172
    • Ludwig, M.1    Reitzner, M.2
  • 63
    • 36549010088 scopus 로고    scopus 로고
    • Limit theorems for the diameter of a random sample in the unit ball
    • Mayer, M. and Molchanov, I. (2007) Limit theorems for the diameter of a random sample in the unit ball. Extremes, 10, 129-150.
    • (2007) Extremes , vol.10 , pp. 129-150
    • Mayer, M.1    Molchanov, I.2
  • 65
    • 0002793629 scopus 로고
    • On the mean width of random polytopes
    • Müller, J. (1989) On the mean width of random polytopes. Probab. Theory Relat. Fields, 82, 33-37.
    • (1989) Probab. Theory Relat. Fields , vol.82 , pp. 33-37
    • Müller, J.1
  • 67
    • 0001840416 scopus 로고
    • Sur l'enveloppe convexe des nuages de points aldéatoires dans - n I
    • Raynaud, H. (1970) Sur l'enveloppe convexe des nuages de points aldéatoires dans - n I. J. Appl. Probab., 7, 35-48.
    • (1970) J. Appl. Probab. , vol.7 , pp. 35-48
    • Raynaud, H.1
  • 68
    • 0036014648 scopus 로고    scopus 로고
    • Random points on the boundary of smooth convex bodies
    • Reitzner, M. (2002) Random points on the boundary of smooth convex bodies. Trans. Amer. Math. Soc., 354, 2243-2278.
    • (2002) Trans. Amer. Math. Soc. , vol.354 , pp. 2243-2278
    • Reitzner, M.1
  • 69
    • 0348222278 scopus 로고    scopus 로고
    • Random polytopes and the Efron-Stein jackknife inequality
    • Reitzner, M. (2003) Random polytopes and the Efron-Stein jackknife inequality. Ann. Probab., 31, 2136-2166.
    • (2003) Ann. Probab. , pp. 2136-2166
    • Reitzner, M.1
  • 70
    • 26844568396 scopus 로고    scopus 로고
    • Stochastical approximation of smooth convex bodies
    • Reitzner, M. (2004) Stochastical approximation of smooth convex bodies. Mathematika, 51, 11-29.
    • (2004) Mathematika , vol.51 , pp. 11-29
    • Reitzner, M.1
  • 71
    • 27844503003 scopus 로고    scopus 로고
    • Central limit theorems for random polytopes
    • Reitzner, M. (2005a). Central limit theorems for random polytopes. Probab. Theory Relat. Fields, 133, 483-507.
    • (2005) Probab. Theory Relat. Fields , vol.133 , pp. 483-507
    • Reitzner, M.1
  • 72
    • 6444228027 scopus 로고    scopus 로고
    • The combinatorial structure of random polytopes
    • Reitzner, M. (2005b). The combinatorial structure of random polytopes. Adv. Math., 191, 178-208.
    • (2005) Adv. Math. , vol.191 , pp. 178-208
    • Reitzner, M.1
  • 73
    • 0001888582 scopus 로고
    • Über die konvexe Hülle von n zufällig gewählten Punkten
    • Rényi, A. and Sulanke, R. (1963) Über die konvexe Hülle von n zufällig gewählten Punkten. Z. Wahrsch. Verw. Geb., 2, 75-84.
    • (1963) Z. Wahrsch. Verw. Geb. , vol.2 , pp. 75-84
    • Rényi, A.1    Sulanke, R.2
  • 74
    • 4243406573 scopus 로고
    • Über die konvexe Hülle von n zufällig gewählten Punkten II
    • Rdényi, A. and Sulanke, R. (1964) Über die konvexe Hülle von n zufällig gewählten Punkten II. Z. Wahrsch. Verw. Geb., 3, 138-147.
    • (1964) Z. Wahrsch. Verw. Geb. , vol.3 , pp. 138-147
    • Rdényi, A.1    Sulanke, R.2
  • 75
    • 34250520636 scopus 로고
    • Zufällige konvexe Polygone in einem Ringge-biet
    • Rdényi, A. and Sulanke, R. (1968) Zufällige konvexe Polygone in einem Ringge-biet. Z. Wahrsch. Verw. Geb., 9, 146-157.
    • (1968) Z. Wahrsch. Verw. Geb. , vol.9 , pp. 146-157
    • Rdényi, A.1    Sulanke, R.2
  • 76
    • 38249041435 scopus 로고
    • Martingale inequalities and the jackknife estimate of variance
    • Rhee, W. T. and Talagrand, M. (1986) Martingale inequalities and the jackknife estimate of variance. Statist. Probab. Lett., 4, 5-6.
    • (1986) Statist. Probab. Lett. , vol.4 , pp. 5-6
    • Rhee, W.T.1    Talagrand, M.2
  • 77
    • 0000060362 scopus 로고
    • On normal approximation rates for certain sums of dependent random variables
    • Rinott, Y. (1994) On normal approximation rates for certain sums of dependent random variables. J. Comput. Appl. Math., 55, 135-143.
    • (1994) J. Comput. Appl. Math. , vol.55 , pp. 135-143
    • Rinott, Y.1
  • 78
    • 33746801607 scopus 로고    scopus 로고
    • Geometric approach to error correcting codes and reconstruction of signals
    • Rudelson, M. and Vershynin, R. (2005) Geometric approach to error correcting codes and reconstruction of signals. Int. Math. Res. Not., 64, 4019-4041.
    • (2005) Int. Math. Res. Not. , vol.64 , pp. 4019-4041
    • Rudelson, M.1    Vershynin, R.2
  • 79
    • 84915617434 scopus 로고
    • Random polytopes generated by anisotropic hyperplanes
    • Schneider, R. (1982) Random polytopes generated by anisotropic hyperplanes. Bull. London Math. Soc., 14, 549-553.
    • (1982) Bull. London Math. Soc. , vol.14 , pp. 549-553
    • Schneider, R.1
  • 80
    • 33646777279 scopus 로고
    • Approximation of convex bodies by random polytopes
    • Schneider, R. (1987) Approximation of convex bodies by random polytopes. Aequationes Math., 32, 304-310.
    • (1987) Aequationes Math. , vol.32 , pp. 304-310
    • Schneider, R.1
  • 81
    • 84986685654 scopus 로고
    • Random approximation of convex sets
    • Schneider, R. (1988) Random approximation of convex sets. J. Microscopy, 151, 211-227.
    • (1988) J. Microscopy , vol.151 , pp. 211-227
    • Schneider, R.1
  • 83
    • 51949090813 scopus 로고    scopus 로고
    • Variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points
    • Schreiber, T., and Yukich, J. E. (2008) Variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points. Ann. Probab., 36, 363-396.
    • (2008) Ann. Probab. , vol.36 , pp. 363-396
    • Schreiber, T.1    Yukich, J.E.2
  • 84
    • 51249172341 scopus 로고
    • The convex floating body and polyhedral approximation
    • Schütt, C. (1991) The convex floating body and polyhedral approximation. Israel J. Math., 73, 65-77.
    • (1991) Israel J. Math. , vol.73 , pp. 65-77
    • Schütt, C.1
  • 85
    • 0001709834 scopus 로고
    • The convex floating body
    • Schütt, C. and Werner, E. (1990) The convex floating body. Math. Scand., 66, 275-290.
    • (1990) Math. Scand. , vol.66 , pp. 275-290
    • Schütt, C.1    Werner, E.2
  • 86
    • 0034311783 scopus 로고    scopus 로고
    • Random polytopes with vertices on the boundary of a convex body
    • Schütt, C. and Werner, E. (2000) Random polytopes with vertices on the boundary of a convex body. C. R. Acad. Sci. Paris I, 331, 1-5.
    • (2000) C. R. Acad. Sci. Paris I , vol.331 , pp. 1-5
    • Schütt, C.1    Werner, E.2
  • 87
    • 84919998797 scopus 로고
    • Regional Conference Series in Applied Mathematics, SIAM, Philadelphia
    • Solomon, H. (1978). Geometric Probability, Volume 28 of Regional Conference Series in Applied Mathematics. SIAM, Philadelphia.
    • (1978) Geometric Probability , vol.28
    • Solomon, H.1
  • 88
    • 0038860738 scopus 로고
    • Asymptotic behaviour of the number of faces of random polyhedra and the neighbourliness problem
    • Vershik, A. M. and Sporyshev, P. V. (1992) Asymptotic behaviour of the number of faces of random polyhedra and the neighbourliness problem. Selecta Math. Soviet., 11, 181-201.
    • (1992) Selecta Math. Soviet. , vol.11 , pp. 181-201
    • Vershik, A.M.1    Sporyshev, P.V.2
  • 89
    • 32344444110 scopus 로고    scopus 로고
    • Sharp concentration of random polytopes
    • Vu, V. H. (2005) Sharp concentration of random polytopes. Geom. Funct. Anal., 15, 1284-1318.
    • (2005) Geom. Funct. Anal. , vol.15 , pp. 1284-1318
    • Vu, V.H.1
  • 90
    • 33748883088 scopus 로고    scopus 로고
    • Central limit theorems for random polytopes in a smooth convex set
    • Vu, V. H. (2006) Central limit theorems for random polytopes in a smooth convex set. Adv. Math., 207, 221-243.
    • (2006) Adv. Math. , vol.207 , pp. 221-243
    • Vu, V.H.1
  • 91
    • 0039894312 scopus 로고    scopus 로고
    • A continuous analogue of the upper bound theorem
    • Wagner, U. and Welzl, E. (2001) A continuous analogue of the upper bound theorem. Discrete Comput. Geom., 26, 205-219.
    • (2001) Discrete Comput. Geom. , vol.26 , pp. 205-219
    • Wagner, U.1    Welzl, E.2
  • 92
    • 0000623082 scopus 로고
    • A problem in geometric probability
    • Wendel, J. G. (1962) A problem in geometric probability. Math. Scand., 11, 109-111.
    • (1962) Math. Scand. , vol.11 , pp. 109-111
    • Wendel, J.G.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.