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Approximation of convex sets by polytopes

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The survey contains results related to different aspects of polyhedral approximation of convex bodies and some adjacent problems.

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References

  1. F. Affentranger and R. Schneider, “Random projections of regular simplexes,” Discr. Comput. Geom., 7, 219–226 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  2. F. Affentranger and J. Wieaker, “On the convex hull of uniform random points in a simple d-polytope,” Discr. Comput. Geom., 6(1991), 291–305 (1991).

    Article  MATH  Google Scholar 

  3. H.-K. Ahn, P. Brass, O. Cheong, H.-S. Na, C.-S. Shin, and A. Vigneron, “Approximation algorithms for inscribing or circumscribing an axially symmetric polygon to a convex polygon,” in: Computing and Combinatorics. Proc. 10th Annual Int. Conf. COCOON 2004, Jeju Island, Korea, August 17–20, 2004, Lect. Notes Comput. Sci., 3106, Springer-Verlag, Berlin (2004), pp. 259–267.

    Google Scholar 

  4. G. E. Andrews, “A lower bound for the volume of strictly convex bodies with many boundary lattice points,” Trans. Amer. Math. Soc., 106, 270–279 (1963).

    Article  MATH  MathSciNet  Google Scholar 

  5. V. I. Arnold, “Statistics of integer-valued convex polygons,” Funkts. Anal. Prilozh., 14, 1–3 (1980).

    Google Scholar 

  6. E. Asplund, “Comparison of plane symmetric convex bodies and parallelograms,” Math. Scand., 8, 171–180 (1960).

    MATH  MathSciNet  Google Scholar 

  7. M. J. Atallah, “A linear time algorithm for the Hausdorff distance between convex polygons,” Inf. Process. Lett., 17, 207–209 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  8. M. J. Atallah, C. C. Ribeiro, and S. Lifschitz, “A linear time algorithm for the computation of some distance functions between convex polygons,” RAIRO, Rech. Oper., 25, 413–424 (1991).

    MATH  MathSciNet  Google Scholar 

  9. I. K. Babenko, “Asymptotic volume of tori and geometry of convex bodies,” Mat. Zametki, 44, 177–190 (1988).

    MATH  MathSciNet  Google Scholar 

  10. A. Balog and I. Bárány, “On the convex hull of the integer points in a disc,” in: Discrete and Computational Geometry. Proc. DIMACS Spec. Year Workshops 1989–90, DIMACS, Ser. Discr. Math. Theor. Comput. Sci., 6, 39–44 (1991).

  11. A. Balog and I. Bárány, “On the convex hull of the integer points in a disk,” Discr. Comput. Geom., 6, 39–44 (1992).

    Google Scholar 

  12. A. Balog and J.-M. Deshorlies, “On some convex lattice polytopes,” in: Number Theory in Progress, de Gruyter (1999), pp. 591–606.

  13. Yu. M. Baryshnikov and R. A. Vitale, “Regular simplices and Gaussian samples,” Discr. Comput. Geom., 11, 141–147 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  14. I. Bárány, “Random polytopes in smooth convex bodies,” Mathematica, 39, 81–92 (1992).

    MATH  Google Scholar 

  15. I. Bárány, “The limit shape of convex lattice polygons,” Discr. Comput. Geom., 13, 270–295 (1995).

    Google Scholar 

  16. I. Bárány, “Affine perimeter and limit shape,” J. Reine Angew. Math., 484, 71–84 (1997).

    MATH  MathSciNet  Google Scholar 

  17. I. Bárány, “Sylvester’s question: The probability that n points are in convex position,” Ann. Probab., 27, 2020–2034 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  18. I. Bárány, “A note on Sylvester’s four-point problem,” Stud. Sci. Math. Hung., 38, 73–77 (2001).

    MATH  Google Scholar 

  19. I. Bárány, “Random points, convex bodies, lattices,” in: Proc. Int. Congr. Math. 2002, 3 (2002), pp. 527–535.

    Google Scholar 

  20. I. Bárány and K. Böröczky, Jr., “Lattice points on the boundary of the integer hull,” in: Discrete Geometry. In honor of W. Kuperberg’s 60th birthday, Pure Appl. Math., Marcel Dekker, 253, (2003), pp. 33–47.

    Google Scholar 

  21. I. Bárány and C. Buchta, “Random polytopes in a convex polytope, independence of shape, and concentration of vertices,” Math. Ann., 297, 467–497 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  22. I. Bárány and L. Dalla, “Few points to generate a random polytope,” Mathematika, 44, 325–331 (1997).

    MATH  MathSciNet  Google Scholar 

  23. I. Bárány and Z. Füredi, “Approximation of the sphere by polytopes having few vertices,” Proc. Am. Math. Soc., 102, 651–659 (1988).

    Article  MATH  Google Scholar 

  24. I. Bárány and Z. Füredi, “On the shape of the convex hull of random points,” Probab. Theory Rel. Fields, 77, 231–240 (1988).

    Article  MATH  Google Scholar 

  25. I. Bárány, R. Howe, and L. Lovász, “On integer points in polyhedra: A lower bound,” Combinatorica, 12, 135–142 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  26. I. Bárány and D. G. Larman, “The convex hull of the integer points in the large ball,” Math. Ann., 312, 167–181 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  27. I. Bárány and D. G. Larman, “Convex bodies, economic cap coverings, random polytopes,” Mathematika, 35, 274–291 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  28. I. Bárány and J. Matousek, “The randomized integer convex hull,” Discr. Comput. Geom., 33, 3–25 (2005).

    Article  MATH  Google Scholar 

  29. I. Bárány and A. Pór, “On 0-1 polytopes with many faces,” Adv. Math., 161, 209–228 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  30. I. Bárány, G. Rote, W. Steiger, and C.-H. Zhang, “A central limit theorem for convex chains in the square,” Discr. Comput. Geom., 23, 35–50 (2000).

    Article  MATH  Google Scholar 

  31. I. Bárány and N. Tokushige, “The minimum area of convex lattice n-gons,” Combinatorica, 24, 171–185 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  32. I. Bárány and A. M. Vershik, “On the number of convex lattice polytopes,” GAFA J., 2, 381–393 (1992).

    Article  MATH  Google Scholar 

  33. A. Bielecki and K. Radziszewski, “Sur les parallélépipèdes inscrits dans les corps convexes,” Ann. Univ. M. Curie-Sclodowska, Sec. A, 7, 97–100 (1954).

    MathSciNet  Google Scholar 

  34. W. Blaschke, Kreis und Kugel, Walter de Gruyter, Berlin (1956).

    MATH  Google Scholar 

  35. W. Blaschke, “Über affine Geometrie, III,” Ber. Verh. Sächs. Ges. Wiss. Leipzig, Math.-Phys., 69, 3–12 (1917).

    Google Scholar 

  36. W. Blaschke, “Über affine Geometrie, IX,” Ber. Verh. Sächs. Ges. Wiss. Leipzig, Math.-Phys., 69, 412–420 (1917).

    Google Scholar 

  37. W. Blaschke, Affine Differenzialgeometrie, Springer-Verlag, 1923.

  38. J. Bokowski, “Konvexe Körper approximierande Polytopclassen,” Elem. Math., 32, 88–90 (1977).

    MATH  MathSciNet  Google Scholar 

  39. J. Bokowski and C. Schulz, “Dichte Klassen konvexer Polytope,” Math. Z., 160, 173–182 (1978).

    Article  MATH  MathSciNet  Google Scholar 

  40. J. Bokowski and P. Mani-Levinska, “Approximation of convex bodies by polytopes with uniformly bounded valences,” Monatsh. Math., 104, 261–264 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  41. W. Boratyński, “Approximation of convex bodies by parallelotopes,” in: Proc. 7th Int. Conf. on Engineering, Computer Graphics, and Descriptive Geometry, Cracow, Poland (1996), pp. 259–260.

  42. K. Böröczky, Jr., “Approximation of general smooth convex bodies,” Adv. Math, 153, 325–341 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  43. K. Böröczky, Jr., “Polytopal approximation bounding the number of k-faces,” J. Approx. Theory, 102, 263–285 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  44. K. Böröczky, Jr., “The error of polytopal approximation with respect to the symmetric difference metric and the L p metric,” Isr. J. Math, 117, 1–28 (2000).

    Article  MATH  Google Scholar 

  45. K. Böröczky, Jr., “About the error term for best approximation with respect to the Hausdorff related metrics,” Discr. Comput. Geom., 25, 293–309 (2001).

    MATH  Google Scholar 

  46. K. Böröczky, Jr. and M. Ludwig, “Approximation of convex bodies and a momentum lemma for power diagram,” Monatsh. Math., 127, 101–110 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  47. K. Böröczky, Jr. and M. Reitzner, “Approximation of smooth convex bodies by random circumscribed polytopes,” Ann. Appl. Probab., 14, 239–273 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  48. J. Bourgain and J. Lindenstrauss, “Distribution of points on sphere and approximation by zonotopes,” Isr. J. Math., 64, 25–31 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  49. J. Bourgain, J. Lindenstrauss, and V. D. Milman, “Approximation of zonoids by zonotopes,” Acta Math., 162, 73–141 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  50. E. M. Bronshtein, “Extremal convex functions,” Sib. Mat. Zh., 19, 10–18 (1978).

    Google Scholar 

  51. E. M. Bronshtein and L. D. Ivanov, “Approximation of convex sets by polyhedrons,” Sib. Mat. Zh., 16, 1110–1112 (1975).

    Google Scholar 

  52. C. Buchta, “Über die konvexe Hölle von Zufallspunkten in Eibereichen,” Elem. Math., 38, 153–156 (1983).

    MATH  MathSciNet  Google Scholar 

  53. C. Buchta, “Stochastische Approximation konvexer Polygone,” Z. Wahrscheinlichkeitstheor. Verw. Geb., 67, 283–304 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  54. C. Buchta, “A note on the volume of a random polytope in a tetrahedron,” Ill. J. Math., 30, 653–659 (1986).

    MATH  MathSciNet  Google Scholar 

  55. C. Buchta, “A remark on random approximation of simple polytopes,” Anz. Österr. Akad. Wiss., Math.-Naturwiss., 2, 17–20 (1989).

    MathSciNet  Google Scholar 

  56. C. Buchta and M. Reitzner, “What is the expected volume of a tetrahedron whose vertices are chosen at random from a given tetrahedron?” Anz. Österr. Akad. Wiss., Math.-Naturwiss., 8, 63–68 (1992).

    MathSciNet  Google Scholar 

  57. C. Buchta and M. Reitzner, “Equiaffine inner parallel curves of a plane convex body and the convex hulls of randomly chosen points,” Probab. Theory Relat. Fields, 108, 385–415 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  58. R. E. Burkard, H. W. Hamacher, and G. Rote, “Sandwich approximation of univariate convex functions with its application to separable convex programming,” Naval Res. Logist., 38, 911–924 (1991).

    MATH  MathSciNet  Google Scholar 

  59. L. V. Burmistrova, On a new method of approximation of convex compact bodies by polyhedrons, Preprint, Comput. Center of the Russian Academy of Sciences (1999).

  60. L. V. Burmistrova, “On a new method of approximation of convex compact bodies by polyhedrons,” Zh. Vychisl. Mat. Mat. Fiz., 40, 1475–1490 (2000).

    MathSciNet  Google Scholar 

  61. V. A. Bushenkov, “Iterative method of construction of optimal projections of convex polyhedron sets,” Zh. Vychisl. Mat. Mat. Fiz., 25, 1285–1292 (1985).

    MATH  MathSciNet  Google Scholar 

  62. V. A. Bushenkov and A. V. Lotov, “Constructions and applications of generalized attainability sets,” Preprint, Comput. Center of the Russian Academy of Sciences (1982).

  63. A. J. Cabo and P. Groeneboom, “Limit theorems for functionals of convex hulls,” Probab. Theory Relat. Fields, 100, 31–55 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  64. G. H. L. Cheang and A. R. Barron, “A better approximation for balls,” J. Approx. Theory, 104, 183–203 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  65. L. Chen, “New analysis of the sphere covering problems and optimal polytope approximation of convex bodies,” J. Approx. Theory, 133, 134–145 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  66. O. L. Chernykh, “On the construction of the convex span of a finite set of points in approximate calculations,” Zh. Vychisl. Mat. Mat. Fiz., 28, 1386–1396 (1988).

    MathSciNet  Google Scholar 

  67. K. L. Clarkson, “Algorithms for polytope covering and approximation,” in: Proc. 3rd Workshop on Algorithms and Data Structures, Lect. Notes Comput. Sci., 709, Springer-Verlag (1993), pp. 246–252.

  68. W. Cook, M. Hartmann, R. Kannan, and C. McDiarmid, “On integer points in polyhedra,” Combinatorica, 12, 27–37 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  69. L. Dalla and D. G. Larman, “Volumes of a random polytope in a convex set,” in: Applied Geometry and Discrete Mathematics, Festschr. 65th Birthday Victor Klee, DIMACS, Ser. Discret. Math. Theor. Comput. Sci., 4 (1991), pp. 175–180.

  70. L. Devroye, “On the oscillation of the expected number of extreme points of a random set,” Stat. Probab. Lett., 11, 281–286 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  71. P. C. Doyle, J. C. Lagarian, and D. Randall, “Self-parking of centrally symmetric convex bodies in ℝ2,” Discr. Comput. Geom., 8, 171–189 (1992).

    Article  MATH  Google Scholar 

  72. A. Dvoretzky, “Some results on convex bodies and Banach spaces,” in: Proc. Int. Symp. Linear Spaces, Jerusalem, 1960, Jerusalem Academic Press; Pergamon, Oxford (1961), pp. 123–160.

    Google Scholar 

  73. A. Dvoretzky and C. A. Rogers, “Absolute and unconditional convergence in normed linear spaces,” Proc. Nac. Acad. Sci. USA, 36, 192–197 (1950).

    Article  MATH  MathSciNet  Google Scholar 

  74. R. M. Dudley, “Metric entropy of some classes of sets with differentiable boundaries,” J. Approx. Theory, 10, 227–236 (1974).

    Article  MATH  MathSciNet  Google Scholar 

  75. R. Dwyer, “On the convex hull of random points in a polytope,” J. Appl. Probab., 25, 688–699 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  76. R. Dwyer and R. Kannan, “Convex hull of randomly chosen points from a polytope. Parallel algorithms and architectures,” in: Proc. Int. Workshop, Suhl/GDR 1987, Math. Res., 38, 16–24 (1987).

  77. M. E. Dyer and A. M. Frieze, “On the complexity of computing the volume of a polyhedron,” SIAM J. Comput., 17, 967–974 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  78. M. Dyer and A. Frieze, “Computing the volume of convex bodies: A case where randomness provably helps,” in: Probabilistic Combinatorics and Its Applications, Short Course, Proc. Symp. Appl. Math., 44 (1991), pp 123–169.

  79. M. E. Dyer, A. Frieze, and R. Kannan, “A random polynomial-time algorithm for approximating the volume of convex bodies,” J. Assoc. Comput. Mach., 38, 1–17 (1991).

    MATH  MathSciNet  Google Scholar 

  80. M. E. Dyer, Z. Fúredi, and C. McDiarmid, “Random volumes in the n-cube,” in: Polyhedral Combinatorics, Proc. Workshop, Morristown/NJ (USA) 1989, DIMACS, Ser. Discrete Math. Theor. Comput. Sci., 1 (1990), pp. 33–38.

  81. M. E. Dyer, Z. Fúredi, and C. McDiarmid, “Volumes spanned by random points in the hypercube,” Random Struct. Algorithms, 3, 91–106 (1992).

    Article  MATH  Google Scholar 

  82. M. Dyer, P. Gritzmann, and A. Hufnagel, “On the complexity of computing mixed volumes,” SIAM J. Comput., 27, 356–400 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  83. S. M. Dzholdybaeva and G. K. Kamenev, “Numerical study of the effectiveness of the algorithm of approximation of convex bodies by polyhedrons,” Zh. Vychisl. Mat. Mat. Fiz., 32, 857–866 (1992).

    MathSciNet  Google Scholar 

  84. R. V. Efremov, “Estimate of the effectiveness of the adaptive approximation algorithm for smooth convex bodies in the two-dimensional case,” Vestn. Mosk. Univ., Ser. 15, 2, 29–32 (2000).

    MathSciNet  Google Scholar 

  85. R. V. Efremov and G. K. Kamenev, “A priori estimate of the asymptotic effectiveness of one class of polyhedral approximation algorithms for convex bodies,” Zh. Vychisl. Mat. Mat. Fiz., 42, 23–32 (2002).

    MATH  MathSciNet  Google Scholar 

  86. R. V. Efremov, G. K. Kamenev, and A. V. Lotov, “On the construction of efficient descriptions of polyhedrons based on the duality theory for convex sets,” Dokl. Ross. Akad. Nauk, 399, 594–596 (2004).

    MathSciNet  Google Scholar 

  87. H. G. Eggleston, “Approximation of plane convex curves, I. Dowker-type theorems,” Proc. London Math. Soc., 7, No. 3, 351–377 (1957).

    Article  MATH  MathSciNet  Google Scholar 

  88. H. G. Eggleston, Convexity, Cambridge (1958).

  89. T. Esterman, “Über der Vektorenbereich eines konvexen Körpers,” Math. Z., 28, 471–475 (1928).

    Article  MathSciNet  Google Scholar 

  90. E. Fabińska and M. Lassak, “Large equilateral triangles inscribed in the unit disk of a Minkowski plane,” Contrib. Algebra Geom., 45, 517–525 (2004).

    MATH  Google Scholar 

  91. U. Faigle, N. Gademann, and W. Kern, “A random polynomial time algorithm for well-routing convex bodies,” Discr. Appl. Math., 58, 117–144 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  92. L. Fejes Tóth, “Approximation by polygons and polyhedra,” Bull. Amer. Math. Soc., 54, 431–438 (1948).

    Article  MATH  MathSciNet  Google Scholar 

  93. L. Fejes Tóth, Lagerungen in der Ebene auf der Kugel und im Raum, Springer-Verlag, Berlin-Göttingen-Heidelberg (1953).

    MATH  Google Scholar 

  94. L. Fejes T’oth, “Approximation of convex domains by polygons,” Stud. Sci. Math. Hung., 15, 133–138 (1980).

    MathSciNet  Google Scholar 

  95. P. Fischer, “Sequential and parallel algorithms for finding a maximum convex polygon,” Comput. Geom., 7, 187–200 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  96. S. Finch and I. Hueter, “Random convex hulls: A variance revisited,” Adv. Appl. Probab., 36, 981–986 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  97. T. Fleiner, V. Kaibel, and G. Rote, “Upper bounds on the maximal number of facets of 0/1-polytopes,” Eur. J. Combinat., 21, 121–130 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  98. A. Florian, “On the perimeter deviation of a convex disk from a polygon,” Rend. Ist. Mat. Univ. Trieste, 24, 177–191 (1992).

    MATH  MathSciNet  Google Scholar 

  99. A. Florian, “Approximation of convex discs by polygons: The perimeter deviation,” Stud. Sci. Math. Hung., 27, 97–118 (1992).

    MATH  MathSciNet  Google Scholar 

  100. E. Floyd, “Real-valued mappings of spheres,” Proc. Amer. Math. Soc., 6, 957–959 (1955).

    Article  MATH  MathSciNet  Google Scholar 

  101. B. Fruhwirth, R. E. Burkard, and G. Rote, “Approximation of convex curves with applications to the minimum cost flow problem,” Eur. J. Oper. Res., 42, 326–338 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  102. C. M. Fulton and S. K. Stein, “Parallelograms inscribed in convex curves,” Amer. Math. Monthly, 67, 257–258 (1960).

    Article  MATH  MathSciNet  Google Scholar 

  103. P. Georgiev, in: Proc 13 Bulgarian Conf. Math. (1984), pp. 289–303.

  104. S. Glasauer and P. M. Gruber, “Asymptotic estimates for best and stepwise approximation of convex bodies, III,” Forum Math., 9, 383–404 (1997).

    MATH  MathSciNet  Google Scholar 

  105. S. Glasauer and R. Schneider, “Asymptotic approximation of smooth convex bodies by polytopes,” Forum Math., 8, 363–377 (1996).

    MATH  MathSciNet  Google Scholar 

  106. A. M. Gleason, “A curvature formula,” Amer. J. Math., 101, 86–93 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  107. E. D. Gluskin, “On the sum of intervals,” in: Geometric Aspects of Functional Analysis. Proc. Israel Semin. (GAFA) 2001–2002, Lect. Notes Math., 1807, Springer-Verlag, Berlin (2003), pp. 122–130.

    Google Scholar 

  108. Y. Gordon, M. Meyer, and S. Reisner, “Volume approximation of convex bodies by polytopes: A constructive method,” Stud. Math., 111, 81–95 (1994).

    MATH  MathSciNet  Google Scholar 

  109. Y. Gordon, M. Meyer, and S. Reisner, “Constructing a polytope to approximate a convex body,” Geom. Dedic., 57, 217–222 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  110. Y. Gordon, S. Reisner, and C. Schütt, “Umbrellas and polytopal approximation of the Euclidean ball,” J. Approx. Theory, 90, 9–22 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  111. P. Gritzmann, V. Klee, and D. Larman, “Largest j-simplices in d-polytopes,” Discr. Comput. Geom., 13, 477–515 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  112. P. Gritzman and M. Lassak, “Estimates for the minimal wight of polytopes inscribed in convex bodies,” Discr. Comput. Geom., 4, 627–635 (1989).

    Article  Google Scholar 

  113. P. Groeneboom, “Limit theorems for convex hulls,” Probab. Theory Relat. Fields, 79, 327–368 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  114. W. Gross, “Über affine Geometrie, XIII,” Ber. Verh. Sächs. Acad. Wis. Leipzig. Math.-Nat, 70, 38–54 (1918).

    Google Scholar 

  115. P. M. Gruber, “In most cases approximation is irregular,” Rend. Sem. Mat. Univers. Torino, 41, 19–33 (1983).

    MATH  MathSciNet  Google Scholar 

  116. P. M. Gruber, “Volume approximation of convex bodies by inscribed polytopes,” Math Ann., 281, 229–245 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  117. P. M. Gruber, “Volume approximation of convex bodies by circumscribed polytopes,” in: Applied Geometry and Discrete Mathematics, DIMACS Ser. 4, Amer. Math. Soc., Providence (1991), pp. 309–317.

    Google Scholar 

  118. P. M. Gruber, “The space of convex bodies,” in: Handbook of Convex Geometry, Elsevier (1993), pp. 301–317.

  119. P. M. Gruber, “Aspects of approximation of convex bodies,” in: Handbook of Convex Geometry, Elsevier (1993), pp. 319–345.

  120. P. M. Gruber, “Asymptotic estimates for best and stepwise approximation of convex bodies, I,” Forum Math., 5, 281–297 (1993).

    MATH  MathSciNet  Google Scholar 

  121. P. M. Gruber, “Asymptotic estimates for best and stepwise approximation of convex bodies, II,” Forum Math., 5, 521–537 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  122. P. M. Gruber, “Approximation by convex polytopes. Polytopes: Abstract, convex and computational,” in: Proc. NATO Adv. Stud. Inst. Scarborough, Ontario, Canada, August 20–September 3, 1993, NATO ASI Ser., Ser. C, Math. Phys. Sci., 440, Kluwer Academic Publishers, Dordrecht (1994), pp. 173–203.

    Google Scholar 

  123. P. M. Gruber, “Comparison of best and random approximation of convex bodies by polytopes,” Rend. Circ. Math. Palermo (2), 50, 189–216 (1997).

    MathSciNet  Google Scholar 

  124. P. M. Gruber, “A short analytic proof of Fejes Tóth’s theorem on sums of moments,” Aequationes Math., 58, 291–295 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  125. P. M. Gruber, “Error of asymptotic formulae for volume approximation of convex bodies in E 3,” Tr. Mat. Inst. Steklova, 239, 106–117 (2002).

    MathSciNet  Google Scholar 

  126. P. M. Gruber, “Error of asymptotic formulae for volume approximation of convex bodies in E 3,” Monatsh. Math., 135, 271–304 (2002).

    MathSciNet  Google Scholar 

  127. P. M. Gruber and P. Kenderov, “Approximation of convex bodies by polytopes,” Rend. Circ. Mat. Palermo (2), 31, 195–225 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  128. B. Grünbaum, Studies in Combinatorial Geometry and the Theory of Convex Bodies [Russian translation], Nauka, Moscow (1971).

    MATH  Google Scholar 

  129. Q. Guo and S. Kajiser, “Approximation of convex bodies by convex bodies,” Northeast Math. J, 19, 323–332 (2003).

    MATH  MathSciNet  Google Scholar 

  130. H. Hadwiger, “Volumschätzung für die eine Eikörper überdeckenden und unterdeckenden Parallelotope,” Elem. Math., 10, 122–124 (1955).

    MATH  MathSciNet  Google Scholar 

  131. H. Hadwiger, Lectures on the Volume, Surface Area, and Isoperimetry [Russian translation], Nauka, Moscow (1966).

    Google Scholar 

  132. A. C. Hayes and D. G. Larman, “The vertices of the knapsack polytope,” Discr. Appl. Math., 6, 135–138 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  133. I. Hueter, “The convex hull of a normal sample,” Adv. Appl. Probab., 26, 855–875 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  134. I. Hueter, “Limit theorems for the convex hull of random points in higher dimensions,” Trans. Amer. Math. Soc., 351, 4337–4363 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  135. D. Hug, G. O. Munsonius, and M. Reitzner, “Asymptotic mean values of Gaussian polytopes,” Beitr. Algebra Geom., 45, 531–548 (2004).

    MATH  MathSciNet  Google Scholar 

  136. D. Hug and M. Reitzner, “Gaussian polytopes: Variances and limit theorems,” Adv. Appl. Probab., 37, 297–320 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  137. S. Johansen, “The extremal convex functions,” Math. Scand., 34, 61–68 (1974).

    MATH  MathSciNet  Google Scholar 

  138. S. Kakutani, “A proof that there exists a circumscribing cube around any bounded closed convex sets in R 3,” Ann. Math., 43, 739–741 (1942).

    Article  MathSciNet  Google Scholar 

  139. G. K. Kamenev, “On one class of adaptive algorithms of approximation of convex bodies by polyhedrons,” Zh. Vychisl. Mat. Mat. Fiz., 32, 136–152 (1992).

    MATH  MathSciNet  Google Scholar 

  140. G. K. Kamenev, “On the effectiveness of Hausdorff algorithms of polyhedral approximations of convex bodies,” Zh. Vychisl. Mat. Mat. Fiz., 33, 796–805 (1993).

    MATH  MathSciNet  Google Scholar 

  141. G. K. Kamenev, “On the examining of one algorithm of approximation of convex bodies,” Zh. Vychisl. Mat. Mat. Fiz., 34, No. 4, 521–528 (1994).

    MathSciNet  Google Scholar 

  142. G. K. Kamenev, “Algorithm of approaching polyhedrons,” Zh. Vychisl. Mat. Mat. Fiz., 36, 134–147 (1996).

    MathSciNet  Google Scholar 

  143. G. K. Kamenev, “Effective algorithms of internal polyhedral approximation of nonsmooth convex bodies,” Zh. Vychisl. Mat. Mat. Fiz., 39, 423–427 (1999).

    MATH  MathSciNet  Google Scholar 

  144. G. K. Kamenev, “On approximative properties of nonsmooth convex disks,” Zh. Vychisl. Mat. Mat. Fiz., 40, 1464–1474 (2000).

    MathSciNet  Google Scholar 

  145. G. K. Kamenev, “On a method of polyhedral approximation of convex bodies that is optimal with respect to the number of calculations of support and distance functions,” Dokl. Ross. Akad. Nauk, 388, 309–311 (2003).

    MATH  MathSciNet  Google Scholar 

  146. G. K. Kamenev, “Self-dual adaptive polyhedral approximation algorithms for convex bodies,” Zh. Vychisl. Mat. Mat. Fiz., 43, 1123–1137 (2003).

    MATH  MathSciNet  Google Scholar 

  147. R. Kannan, L. Lovsz, and M. Simonovits, “Random walks and an O*(n 5) volume algorithm for convex bodies,” Random Struct. Algorithms, 11, 1–50 (1997).

    Article  MATH  Google Scholar 

  148. B. S. Kashin, “On parallelepipeds of minimal volume that contain a convex body,” Mat. Zametki, 45, 134–135 (1989).

    MATH  MathSciNet  Google Scholar 

  149. P. Kenderov, “Approximation of plane convex compacta by polygons,” Dokl. Bulg. Akad. Nauk, 33, 889–891 (1980).

    MATH  MathSciNet  Google Scholar 

  150. P. Kenderov, “Polygonal approximation of plane convex compacta,” J. Approx. Theory, 38, 221–239 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  151. K.-H. Küfer, “On the approximation of a ball by random polytopes,” Adv. Appl. Probab., 26, 876–892 (1994).

    Article  MATH  Google Scholar 

  152. V. Klee, “Facet-centroids and volume minimization,” Stud. Sci. Math. Hungar., 21, 143–147 (1986).

    MATH  MathSciNet  Google Scholar 

  153. V. Klee and M. C. Laskowski, “Finding the smallest triangle containing a given convex polygon,” J. Algorithms, 6, 359–366 (1985).

    Article  MATH  MathSciNet  Google Scholar 

  154. M. Kochol, “A note on approximation of a ball by polytopes,” Discr. Optim., 1, 229–231 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  155. A. Kosinński, “A proof of an Auerbach-Banach-Mazur-Ulam theorem on convex bodies,” Colloq. Math., 4, 216–218 (1957).

    MathSciNet  Google Scholar 

  156. D. Kylzow, A. Kuba, and A. Volcic, “An algorithm for reconstructing convex bodies from their projections,” Discr. Comput. Geom., 4, 205–237 (1989).

    Article  Google Scholar 

  157. Z. Lángi, “On seven points on the boundary of a plane convex body in large relative distances,” Contrib. Alg. Geom., 45, 275–281 (2004).

    MATH  Google Scholar 

  158. M. Lassak, “Approximation of convex bodies by parallelotopes,” Bull. Pol. Acad. Sci., Math., 39, 219–233 (1991).

    MathSciNet  MATH  Google Scholar 

  159. M. Lassak, “On five points in a plane convex body pairwise in at least unit relative distances,” Coll. Math. Soc. János Bolyai, Szegel, 69, 245–247 (1991).

    Google Scholar 

  160. M. Lassak, “Approximation of convex bodies by triangles,” Proc. Amer. Math. Soc, 115, 207–210 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  161. M. Lassak, “Approximation of convex bodies by rectangles,” Geom. Dedic., 47, 111–117 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  162. M. Lassak, “Relationship between widths of a convex body and of an inscribed parallelotope,” Bull. Austr. Math. Soc., 63, 133–140 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  163. M. Lassak, “Affine-regular hexagons of extreme areas inscribed in a centrally symmetric convex body,” Adv. Geom., 3, 45–51 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  164. M. Lassak, “On relatively equilateral polygons inscribed in a convex body,” Publ. Math. Debrecen, 65, 133–148 (2004).

    MATH  MathSciNet  Google Scholar 

  165. F. W. Levi, “DeÜber zwei Sätze von Herrn Besicovitch,” Arch. Math., 3, 125–129 (1952).

    Article  MATH  MathSciNet  Google Scholar 

  166. M. A. Lopez and S. Reisner, “Efficient approximation of convex polygons,” Int. J. Comput. Geom. Appl., 10, 445–452 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  167. M. A. Lopez and S. Reisner, “Linear time approximation of 3D convex polytopes,” Comput. Geom., 23, 291–301 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  168. L. Lovász and M. Simonovits, “Random walks in a convex body and an improved volume algorithm,” Random Struct. Algorithms, 4, 359–412 (1993).

    Article  MATH  Google Scholar 

  169. M. Ludwig, Asymptotische Approximation Konvexer Körper, Ph.D. Thesis, Techn. Univ. Vienna (1992).

  170. M. Ludwig, “Asymptotic approximation of convex curves,” Arch. Math., 63, 377–384 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  171. M. Ludwig, “Asymptotic approximation of convex curves: The Hausdorff metric case,” Arch. Math., 70, 331–336 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  172. M. Ludwig, “A characterization of affine length and asymptotic approximation of convex discs,” Abh. Math. Semin. Univ. Hamb., 69, 75–88 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  173. M. Ludwig, “Asymptotic approximation of smooth convex bodies by general polytopes,” Mathematica, 46, 103–125 (1999).

    MATH  MathSciNet  Google Scholar 

  174. A. M. Macbeath, “An extremal property of the hypersphere,” Proc. Cambrige Phil. Soc., 47, 245–247 (1951).

    MATH  MathSciNet  Google Scholar 

  175. A. M. Macbeath, “Compactness theorem for affine equivalence classes of convex regions,” Can. J. Math., 3, 54–61 (1951).

    MATH  MathSciNet  Google Scholar 

  176. V. V. Makeev, “Inscribed and circumscribed polyhedra of convex bodies and related problems,” Mat. Zametki, 51, 67–71 (1992).

    MathSciNet  Google Scholar 

  177. V. V. Makeev, “Inscribed and circumscribed polyhedra of convex bodies,” Mat. Zametki, 55, 128–130 (1994).

    MathSciNet  Google Scholar 

  178. V. V. Makeev, “On approximation of planar sections of convex bodies,” Zap. Nauch. Semin. POMI, 246, 174–183 (1997).

    Google Scholar 

  179. V. V. Makeev, “Three-dimensional inscribed and circumscribed polyhedra for convex compact sets,” Algebra Analiz, 12, 1–15 (2000).

    MathSciNet  Google Scholar 

  180. V. V. Makeev, “Three-dimensional inscribed and circumscribed polyhedra for convex compact sets, 2,” Algebra Analiz, 13, 110–133 (2001).

    MathSciNet  Google Scholar 

  181. V. V. Makeev, “On some combinatorial-geometric problems for vector bundles,” Algebra Analiz, 14, 169–191 (2002).

    MATH  MathSciNet  Google Scholar 

  182. V. V. Makeev and A. S. Mukhin, “On the existence of a common section for several convex compact sets that has prescribed properties,” Zap. Nauch. Semin. POMI, 261, 198–203 (1999).

    Google Scholar 

  183. B. Massü, “On the LLN for the number of vertices of a random convex hull,” Adv. Appl. Probab., 32, 675–681 (2000).

    Article  Google Scholar 

  184. D. E. McClure and R. A. Vitalie, “Polygonal approximation of plane convex bodies,” J. Math. Anal. Appl., 51, 326–358 (1975).

    Article  MATH  MathSciNet  Google Scholar 

  185. J. S. Müller, “Approximation of a ball by random polytopes,” J. Approx. Theory, 63, 198–209 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  186. H. Niederreiter, “On a measure of denseness for sequences,” in: Topics in Classical Number Theory, Colloq. Math. Soc. Janós Bolyai, 34 (1984), pp. 1163–1208.

  187. P. Novotný, “Approximation of convex sets by simplexes,” Geom. Dedic., 50, 53–55 (1994).

    Article  MATH  Google Scholar 

  188. J. O’Rourke, “Finding minimal enclosing boxes,” Int. J. Comput. Inform. Sci., 14, 183–199 (1985).

    Article  MATH  MathSciNet  Google Scholar 

  189. J. O’Rourke, A. Aggarwal, S. Meddila, and M. Baldwin, “An optimal algorithm for finding minimal enclosing triangle,” J. Algorithms, 7, 258–269 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  190. A. Packer, “Polynomial-time approximation of largest simplices in V-polytopes,” Discr. Appl. Math., 134, 213–237 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  191. O. Palmon, “The only convex body with extremal distance from the ball is the simplex,” Isr. J. Math., 80, 337–349 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  192. A. Pelczynski and S. J. Szarek, “On parallelepipeds of minimal volume containing a convex symmetric body in ℝn,” Math. Proc. Camb. Phil. Soc., 109, 125–148 (1991).

    MATH  MathSciNet  Google Scholar 

  193. C. M. Petty, “On the geometry of the Minkowski plane,” Riv. Mat. Univ. Parma, 6, 269–292 (1955).

    MATH  MathSciNet  Google Scholar 

  194. V. A. Popov, “Approximation of convex sets,” Izv. Mat. Inst. Bulg. Akad. Nauk, 11, 67–80 (1970).

    MATH  Google Scholar 

  195. K. Radziszewski, “Sur un problème extremal relatif aux figures inscrites et circonscrite aux figures convexes,” Ann. Univ. M. Curie-Sclodowska, Sec. A, 6, 5–18 (1952).

    MathSciNet  Google Scholar 

  196. S. Reisner, C. Schütt, E. Werner, “Dropping a vertex or a facet from a convex polytope,” Forum Math., 13, 359–378 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  197. M. Reitzner, “Inequalities for convex hulls of random points,” Monatsh. Math., 131, 71–78 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  198. M. Reitzner, “Stochastical approximation of smooth convex bodies,” in: Proc. Conf. “Konvexgeometrie,” Oberwolfach, Apr. 22–28 2001, Tagungsber. Math. Forcshungsist. Oberwolfach, 18 (2001), p. 14.

  199. M. Reitzner, “Random points on the boundary of smooth convex bodies,” Trans. Amer. Math. Soc., 354, 2243–2278 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  200. M. Reitzner, “Random polytopes and the Efron-Stein jackknife inequality,” Ann. Probab., 31, 2136–2166 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  201. M. Reitzner, “The combinatorial structure of random polytopes,” Adv. Math., 191, 178–208 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  202. A. Rényi, R. Sulanke, “:Uber die konvexe Hülle von n zufällig gewahlten Punkten,” Z. Wahrscheinlichkeitsth, 2, 75–84 (1963).

    Article  MATH  Google Scholar 

  203. G. Rote, “The convergence rate for the Sandwich algorithm for approximating convex figures in the plane,” in: Proc. 2nd Canad. Conf. on Comput. Geometry, Ottava (1990), pp. 287–290.

  204. G. Rote, “The convergence rate of the Sandwich algorithm for approximating convex functions,” Computing, 48, 337–361 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  205. V. V. Ryabchenko, S. I. Lyashko, O. N. Grushets’kyj, and B. V. Rublyov, “A new algorithm of linear complexity for finding triangles of maximal area inscribed in a convex polygon,” Visn. Kiev. Univ. Im. Tarasa Shevchenka, Ser. Fiz.-Mat. Nauky, 2, 210–214 (2003).

    Google Scholar 

  206. E. Sas, “Ber eine Extremaleigenschalf der Ellipsen,” Compos. Math., 6, 468–470 (1939).

    MATH  MathSciNet  Google Scholar 

  207. G. C. Shephard, “Decomposable convex polyhedra,” Mathematica, 10, 89–95 (1963).

    MATH  MathSciNet  Google Scholar 

  208. R. Schneider, “Zwei Extremalaufgaben für konvexer Bereiche,” Acta Math. Acad. Sci. Hung., 22, 379–383 (1971).

    Article  Google Scholar 

  209. R. Schneider, “Zur optimalen Approximation konvexer Hyperflachen durch Polyeder,” Math. Ann., 256, 289–301 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  210. R. Schneider, “Affine-invariant approximation by convex polytopes,” Studia Sci. Math. Hung., 21, 401–408 (1986).

    MATH  Google Scholar 

  211. R. Schneider, “Polyhedral approximation of smooth convex bodies,” J. Math. Anal. Appl., 128, 470–474 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  212. R. Schneider, “Random approximation of convex sets,” J. Microscopy, 151, 211–227 (1988).

    Google Scholar 

  213. R. Schneider and J. A. Wieacker, “Approximation of convex bodies by polytopes,” Bull. London Math. Soc., 13, 149–156 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  214. C. Schütt, “The convex floating body and polyhedral approximation,” Isr. J. Math., 73, 65–77 (1991).

    Article  MATH  Google Scholar 

  215. O. Schwarzkopf, U. Futchs, G. Rote, and E. Welzl, “Approximation of convex figures by pairs of rectangles,” Comput. Geom., 10, 77–87 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  216. V. N. Shevchenko, Qualitative Problems of Integer-Valued Linear Programming [in Russian], Fizmatlit, Nauka (1995).

  217. Ya. G. Sinai, “Probabilistic approach to the analysis of statistics of convex broken lines,” Funkts. Anal. Prilozh., 28, 16–25 (1994).

    MathSciNet  Google Scholar 

  218. G. Sonnevand, “An optimal sequential algorithm for the uniform approximation of convex functions on [0,1]2,” Appl. Math. Optim., 10, 127–142 (1983).

    Article  MathSciNet  Google Scholar 

  219. W. Süss, “Über Parallelogramme und Rechtecke, die sich ebenen Eibereichen einschreben lassen,” Rend. Math. Appl., 14, 338–341 (1955).

    Google Scholar 

  220. S. Tabachnikov, “On the dual billiard problem,” Adv. Math., 115, 221–249 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  221. P. Valtr, “Probability that n random points are in convex position,” Discr. Comput. Geom. 13, 637–643 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  222. P. Valtr, “The probability that n random points in a triangle are in convex position,” Combinatorica, 16, 567–573 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  223. A. M. Vershik, “Limit forms of convex integer-valued polygons and related topics,” Funkts. Anal. Prilozh., 28, 16–25 (1994).

    MathSciNet  Google Scholar 

  224. A. M. Vershik, “Limit forms of typical geometric configurations and their applications,” Zap. Nauch. Semin. POMI, 280, 73–100 (2001).

    Google Scholar 

  225. A. M. Vershik and P. V. Sporyshev, “Asymptotic behavior of the number of faces of random polyhedra and the neighborliness problem,” Sel. Math. Sov., 11, 181–201 (1992).

    MATH  MathSciNet  Google Scholar 

  226. A. M. Vershik and O. Zeitouni, “Large deviations in the geometry of complex lattice polygons,” Isr. J. Math., 109, 13–27 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  227. J. A. Wieaker, “Eintre Probleme der polyedrishen Approximation,” Diplomarbeit, Univ. Freiburg (1978).

  228. N. V. Zhivkov, “Plane polygonal approximation of bounded convex sets,” Dokl. Bulg. Akad. Nauk, 35, 1631–1634 (1982).

    MathSciNet  Google Scholar 

  229. Y. Zhou and S. Suri, “Algorithms for minimum volume enclosing a simplex in three dimensions,” SIAM J. Comput., 31, No. 5, 1339–1357 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  230. G. M. Ziegler, “Lectures on 0/1 polytopes,” in: Polytopes: Combinatorics and Computation, DMV-Seminars, Birkhäuser (2000), pp. 1–44.

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    E. M. Bronstein

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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 22, Geometry, 2007.

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Bronstein, E.M. Approximation of convex sets by polytopes. J Math Sci 153, 727–762 (2008). https://doi.org/10.1007/s10958-008-9144-x

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