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On the Complexity of Some Basic Problems in Computational Convexity

II. Volume and mixed volumes

  • Chapter
Polytopes: Abstract, Convex and Computational

Part of the book series: NATO ASI Series ((ASIC,volume 440))

Abstract

This paper is the second part of a broader survey of computational convexity, an area of mathematics that has crystallized around a variety of results, problems and applications involving interactions among convex geometry, mathematical programming and computer science. The first part [GrK94a] discussed containment problems. This second part is concerned with computing volumes and mixed volumes of convex polytopes and more general convex bodies. In order to keep the paper self-contained we repeat some aspects that have already been mentioned in [GrK94a]. However, this overlap is limited to Section 1. For further background material and references, see [GrK94a], and for other parts of the survey see [GrK94b] and [GrK94c].

Research of the authors was supported in part by the Deutsche Forschungsgemeinschaft and in part by a joint Max-Planck Research Award. Research of the second author was also supported in part by the National Science Foundation, U.S.A.

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References

  1. Aleksandrov A.D., On the theory of mixed volumes of convex bodies, II. New inequalities between mixed volumes and their applications (in Russian), Math. Sb. N.S. 2 (1937), pp. 1205–1238

    Google Scholar 

  2. Aleksandrov A.D., On the theory of mixed volumes of convex bodies, IV. Mixed discriminants and mixed volumes (in Russian), Math. Sb. N.S. 3 (1938), pp. 227–251

    Google Scholar 

  3. Alexander R., The width and diameter of a simplex, Geometriae Dedicata 6 (1977), pp. 87–94

    Article  MathSciNet  MATH  Google Scholar 

  4. Allgower E.L. and Schmidt P.M., Computing volumes of polyhedra, Math, of Comput. 46 (1986), pp. 171–174

    Article  MathSciNet  MATH  Google Scholar 

  5. Applegate D. and Kannan R., Sampling and integration of near log-concave functions, Proc. 23rd ACM Symp. Th. of Comput. (1990), pp. 156–163

    Google Scholar 

  6. Aurenhammer F., Improved algorithms for disks and balls using power diagrams, J. Algorithms 9 (1988), pp. 151–161

    Article  MathSciNet  MATH  Google Scholar 

  7. Avis D., Bhattacharya B.K. and Imai H., Computing the volume of the union of spheres, The Visual Computer 3 (1988), pp. 323–328

    Article  MATH  Google Scholar 

  8. Avis D. and Fukuda K., A pivoting algorithm for convex hulls and vertex enumeration of arrangements of polyhedra, Proc. 7th Ann. Symp. Comput. Geom., June, 1991, pp. 98–104; Discrete Comput. Geom. 8 (1992), pp. 295-313

    Google Scholar 

  9. Balakrishnan A.V., A contribution to the sphere-packing problem of communication theory, J. Math. Anal. Appl. 3 (1961), pp. 485–506

    Article  MathSciNet  MATH  Google Scholar 

  10. Balakrishnan A.V., Research Problem No. 9: Geometry, Bull. Amer. Math. Soc. 69 (1963), pp. 737–738

    Article  MathSciNet  Google Scholar 

  11. Balakrishnan A.V., Signal selection for space communication chan-nels; In: Advances in Communication Systems, (ed. by A.V. Balakrishnan), Academic Press, New York, 1965, pp, 1–31

    Google Scholar 

  12. Banach S. and Tarski A., Sur la decomposition des ensembles de points en parties respectivement congruents, Fund. Math. 6 (1924), pp. 244–277

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Bárány I. and Füredi Z., Computing the volume is difficult, Proc. ACM Symp. Th. Comp. 8 (1986), pp. 442–447; Discrete Comput. Geom. 2 (1987), pp. 319-326

    Google Scholar 

  15. Barrow D.L. and Smith P.W., Spline notation applied to a volume problem, Amer. Math. Monthly 86 (1979), pp. 50–51

    Article  MathSciNet  MATH  Google Scholar 

  16. Barvinok A.I., Calculation of exponential integrals (in Russian), Zap. Nauchn. Sem. LOMI, TeoriyaSlozhnosti Vychislenii 192 (1991), pp. 149–163

    MATH  Google Scholar 

  17. Barvinok A.I., Computing the volume, counting integral points, and exponential sums, Discrete Comput. Geom. 10 (1993), pp. 123–141

    Article  MathSciNet  MATH  Google Scholar 

  18. Barvinok A.I., A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed, Preprint (to appear)

    Google Scholar 

  19. Bayer M and Lee C., Combinatorial aspects of convex polytopes; In: Handbook of Convex Geometry, (ed. by P. Gruber and J. Wills), North-Holland, Amsterdam, 1993, pp. 251–305

    Google Scholar 

  20. Ben-Israel A.,. A volume associated with m by n matrices, Lin.Alg.Appl., 167 (1992), pp. 87–111

    Article  MathSciNet  MATH  Google Scholar 

  21. Bérad P., Besson G. and Gallot A.S., Sur une inégalité isopérimétrique qui géneralise celle de Paul Levy — Gromov, Invent. Math. 80 (1985), pp. 295–308

    Article  MathSciNet  Google Scholar 

  22. Berezin I.S. and Zhidkov N.P., Computing Methods, Vol.1, Pergamon Press, Oxford, 1965

    MATH  Google Scholar 

  23. Berman J. and Hanes K., Volumes of polyhedra inscribed in the unit sphere in E 3, Math. Ann. 188 (1970), pp. 78–84

    Article  MathSciNet  MATH  Google Scholar 

  24. Bernshtein D.N., The number of roots of a system of equations, Funct. Anal. Appl. 9 (1975), pp. 183–185

    Google Scholar 

  25. Betke U., Mixed volumes of polytopes, Arch. Math. 58 (1992), pp. 388–391

    Article  MathSciNet  MATH  Google Scholar 

  26. Betke U. and Henk M., Approximating the volume of convex bodies, Discrete Comput. Geom. 10 (1993), pp. 15–21

    Article  MathSciNet  MATH  Google Scholar 

  27. Betke U. and McMullen P., Estimating the size of convex bodies from projections, J. London Math. Soc. (2) 27 (1983), pp. 525–538

    Article  MathSciNet  MATH  Google Scholar 

  28. Bieri H. and Nef W., A sweep-plane algorithm for computing the volume of polyhedra represented in boolean form, Lin. Alg. Appl. 52/53 (1983), pp. 69–97

    MathSciNet  Google Scholar 

  29. Björner A., Las Vergnas M., Sturmfels B., White N. and Ziegler G., Oriented Matroids, Cambridge University Press, Cambridge, 1993

    MATH  Google Scholar 

  30. Blum A.L. and Kannan R., Learning an intersection of k halfspaces over a uniform distribution, Proc. 34th Symp. Found. Comput. Sci. (1993), pp. 312–320

    Google Scholar 

  31. Blumenthal L.M., Distance Geometry, Oxford Univ. Press, London, 1953

    MATH  Google Scholar 

  32. Blumenthal L.M. and Gillam B.E., Distribution of points in n-space, Amer. Math. Monthly 50 (1943), pp. 181–185

    Article  MathSciNet  MATH  Google Scholar 

  33. Bodlaender H.L., Gritzmann P., Klee V. and Van Leeuwen J., The computational complexity of norm-maximization, Combinatorica 10 (1990), pp. 203–225

    Article  MathSciNet  MATH  Google Scholar 

  34. Bokowski J., Hadwiger H. and Willis J.M., Eine Ungleichung zwis-chen Volumen, Oberfläche und Gitterpunktanzahl konvexer Körper im n-dimensionalen euklidischen Raum, Math. Z. 127 (1972), pp. 363–364

    Article  MathSciNet  MATH  Google Scholar 

  35. Boltyanskii V.G., HilberV’s Third Problem, (trans. by R. Silverman), Winston, Washington, D.C., 1978

    Google Scholar 

  36. Bonnesen T. and Fenchel W., Theorie der konvexen Körper Springer, Berlin, 1934; (reprinted: Chelsea, New York), 1948; The-ory of Convex Bodies, (English edition), BCS Associates, Moscow, Idaho, U.S.A., 1987

    Book  MATH  Google Scholar 

  37. de Boor C. and Hollig K., Recurrence relations for multivariate B-splines, Proc. Amer. Math. Soc. 85 (1982), pp. 397–400

    Article  MathSciNet  MATH  Google Scholar 

  38. Brightwell G. and Winkler P., Counting linear extensions, Order 1991, pp. 225–242

    Google Scholar 

  39. Brion M., Points entiers dans les polyèdres convexes, Ann. Sci.Ècole Norm. Sup. (4) 21 (1988), pp. 653–663

    MathSciNet  MATH  Google Scholar 

  40. BrØndsted A., An Introduction to Convex Polytopes, Springer, New York, 1983

    Book  Google Scholar 

  41. Buchta C. and Reitzinger 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. (1992), pp. 63–68

    Google Scholar 

  42. Burago Y.D. and Zalgaller V.A., Geometric inequalities, Springer, Berlin, 1988

    MATH  Google Scholar 

  43. Burger T., Gritzmann P. and Klee V., Finding optimal shadows of polytopes, in preparation

    Google Scholar 

  44. Burger T., Gritzmann P. and Klee V., Optimizing sections of polytopes, in preparation

    Google Scholar 

  45. Canny J. and Rojas J.M., An optmality condition for determining the exact number of roots of a polynomial system, (to appear)

    Google Scholar 

  46. Chang C.C. and Keisler H.J., Model Theory, North-Holland, Amsterdam, 1973

    MATH  Google Scholar 

  47. Chakerian D.G. and Klamkin M.S., Minimum triangles inscribed in a convex curve, Math. Mag. 46 (1973), pp. 256–260

    Article  MathSciNet  MATH  Google Scholar 

  48. Chazelle B., An optimal convex hull algorithm in any fixed dimension, Discrete Comput. Geom. 10 (1993), pp. 377–409

    Article  MathSciNet  MATH  Google Scholar 

  49. Chen P.-C., Hansen P. and Jaumard B., Partial pivoting in vertex enumeration, RUTCOR Research Report #10-92 (1992)

    Google Scholar 

  50. Chung K.C. and Yao T.H., On lattices admitting unique Lagrange interpolation, SIAM J. Numer. Analysis 14 (1977), pp. 735–743

    Article  MathSciNet  MATH  Google Scholar 

  51. Cohen J. and Hickey T., Two algorithms for determining volumes of convex polyhedra. J Assoc. Comp. Mach. 26 (1979), pp. 401–414

    Article  MathSciNet  MATH  Google Scholar 

  52. Croft H.T., Falconer K.J. and Guy R.K., Unsolved Problems in Geometry, Springer, New York, 1991

    Book  MATH  Google Scholar 

  53. Dantzig G.B., Linear Programming and Extensions, Princeton University Press, Princeton, 1963

    MATH  Google Scholar 

  54. Davis P.J. and Rabinovitz P., Methods of Numerical Integration, (2nd ed.) Academic Press, Orlando, 1984

    MATH  Google Scholar 

  55. Dehn M., Über raumgleiche Polyeder, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. (1900), pp. 345–354

    Google Scholar 

  56. Dörrie H., 100 Great Problems of Elementary Mathematics, Dover, New York, 1965

    MATH  Google Scholar 

  57. Dubins L., Hirsch M. and Karush J., Scissor congruence, Israel J. Math. 1 (1963), pp. 239–247

    Article  MathSciNet  Google Scholar 

  58. Dyer M.E., The complexity of vertex enumeration methods, Math. Oper. Res. 8 (1983), pp. 381–402

    Article  MathSciNet  MATH  Google Scholar 

  59. Dyer M.E., On counting lattice points in polyhedra, SIAM J. Corn-put. 20 (1991), pp. 695–707

    Article  MathSciNet  MATH  Google Scholar 

  60. Dyer M.E. and Frieze A.M., The complexity of computing the volume of a polyhedron, SIAM J. Comput. 17 (1988), pp. 967–974

    Article  MathSciNet  MATH  Google Scholar 

  61. Dyer M.E. and Frieze A.M., Computing the volume of convex bodies: a case where randomness provably helps In: Probabilistic Combinatorics and its Applications, (ed. by Béla Bollobás), Proceedings of Symposia in Applied Mathematics Vol. 44, American Mathematical Society, 1991, pp. 123–169

    Google Scholar 

  62. Dyer M.E., Frieze A.M. and Kannan R., A random polynomial time algorithm for estimating volumes of convex bodies, Proc. 21st Symp. Th. Comput. (1989), pp. 375–381

    Google Scholar 

  63. Dyer M.E., Frieze A.M. and Kannan R., A random polynomial time algorithm for approximating the volumes of convex bodies, J. Assoc. Comp. Mach. (1989), pp. 1–17

    Google Scholar 

  64. Dyer M.E., Gritzmann P. and Hufnagel A., On the complexity of computing (mixed) volumes, manuscript (1994)

    Google Scholar 

  65. Dyer M.E. and Kannan R., On Barvinok’s algorithm for counting lattice points in fixed dimension, manuscript (1993)

    Google Scholar 

  66. Edelsbrunner H., Algorithms in Combinatorial Geometry, Springer, New York etc., 1987

    Book  MATH  Google Scholar 

  67. Edelsbrunner H., Computational geometry; In: Handbook of Con-vex Geometry A, (ed. by P. Gruber and J. Wills), North-Holland, Amsterdam, 1993, pp. 699–735

    Google Scholar 

  68. Edelsbrunner H., The union of balls and its dual shape, Proc. 9th Ann. Sympos. Comput. Geom. (1993), pp. ??-??

    Google Scholar 

  69. Edelsbrunner H. and Fu P., Measuring spacefilling diagrams; (Rept. 1010) Nat. Center Supercomputer Appl., Univ. Illinois, Urbana, Illinois, 1993

    Google Scholar 

  70. Edelsbrunner H. and Mücke P., Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms, ACM Trans. Graphics 9 (1990), pp. 66–104

    Article  MATH  Google Scholar 

  71. Edelsbrunner H., O’Rourke J. and Seidel R., Constructing arrangements of lines and hyperplanes with applications, SIAM J. Computing 15 (1986), pp. 341–363

    Article  MathSciNet  MATH  Google Scholar 

  72. Edelsbrunner H., Seidel R. and Sharir M., On the zone theorem for hyperplane arrangements; In: New Results and Trends in Computer Science, (ed. by H. Maurer), Springer Lecture Notes in Computer Science 555, Berlin, 1991, pp. 108–123; SIAM J. Comput. 22 (1993), pp. 418-429

    Google Scholar 

  73. Edmonds J., Submodular functions, matroids, and certain polyhedra; In: Combinatorial Structures and their Applications, (ed. by R. Guy, H. Hanani, N. Sauer and J. Schönheim), Gordon and Breach, New York, 1970, pp. 69–87

    Google Scholar 

  74. Efrat A., Lindenbaum M. and Sharir M., On finding maximally consistent sets of half spaces; In: Proc. 5th Canad. Confi Comput. Geom., Univ. of Waterloo, Waterloo, Canada, 1993, pp. 432–436

    Google Scholar 

  75. Egorychev G.P., The solution of van der Waerden’s problem for per-manents, Advances in Math. 42 (1981), pp. 299–305

    Article  MathSciNet  MATH  Google Scholar 

  76. Ehrhart E., Sur un probléme de géometrie diophantienne linéaire, J. reine angew. Math. 226 (1967), pp. 1–29; 227 pp. 25-49

    MathSciNet  MATH  Google Scholar 

  77. Ehrhart E., Démonstration de la loi de réciprocité, C.R. Acad. Sci. Paris 265 (1968), pp. 5–9, 91-94

    MathSciNet  Google Scholar 

  78. Ehrhart E., Démonstration de la lot de réciprocité, C.R. Acad, Sci. Paris 266 (1969), pp. 696–697

    MathSciNet  Google Scholar 

  79. Ehrhart E., Polynômes arithmétiques et méthode des polyédres en combinatoire, Birkhäuser, Basel, 1977

    MATH  Google Scholar 

  80. Elekes G., A geometric inequality and the complexity of computing volume, Discrete Comput. Geom. 1 (1986), pp. 289–292

    Article  MathSciNet  MATH  Google Scholar 

  81. Erickson J. and Seidel R., Better lower bounds on detecting affine and spherical degeneracies, Proc. 34th Ann. IEEE Sympos. Found. Comput. Sci. (FOCS93), (1993), pp. 528–536

    Google Scholar 

  82. Falikman D.I., A proof of the van der Waerden conjecture on the permanent of a doubly stochastic matrix, Mat. Zametki 29 (1981), pp. 931–938; English translation: Math. Notes, Acad. Sci. USSR 29 (1981), pp. 475-479

    MathSciNet  Google Scholar 

  83. Farber S.M., On the signal selection problem for phase coherent and incoherent communication channels, Tech. Report No. 4, Communications Theory Lab., Dept. of Electrical Engineering, Calif. Inst. Tech. (1968)

    Google Scholar 

  84. Fejes Tóth L., Lagerungen in der Ebene,auf der Kugel,und im Raum, Springer, Berlin, 1953

    MATH  Google Scholar 

  85. Fejes Tóth L., Regular Figures, Pergamon, Oxford, 1964

    MATH  Google Scholar 

  86. Fenchel W., Inégalités quadratique entre les volumes mixtes des corps convexes, C.R. Acad. Sci. Paris 203 (1936), pp. 647–650

    Google Scholar 

  87. Filliman P., Exterior algebra and projections of polytopes, Discrete Comput. Geom. 5 (1990), pp. 305–322

    Article  MathSciNet  MATH  Google Scholar 

  88. Filliman P., The extreme projections of the regular simplex, Trans. Amer. Math. Soc. 317 (1990), pp. 611–629

    Article  MathSciNet  MATH  Google Scholar 

  89. Filliman P., Volumes of duals and section of polytopes, Mathematika 39 (1992), pp. 67–80

    Article  MathSciNet  MATH  Google Scholar 

  90. Firey W.J., A functional characterization of certain mixed volumes, Israel J. Math. 24 (1976), pp. 274–281

    Article  MathSciNet  MATH  Google Scholar 

  91. Fredman M.L. and Weide B., On the complexity of computing the measure of ∪[a i,b i], Comm. Assoc. Comp. Mach. 21 (1978) pp. 540–544

    MathSciNet  MATH  Google Scholar 

  92. Gajentaan A. and Overmars H., n 2-hard problems in computational geometry, RUU-CS-93-15 (1993), Dept. of Comp. Sci., Univ. of Utrecht, Utrecht, Netherlands

    Google Scholar 

  93. Gardner R.J. and Gritzmann P., Successive determination and verifications of polytopes by their X-rays, J. London Math. Soc. (1994), (to appear)

    Google Scholar 

  94. Gardner R.J. and Wagon S., At long last, the circle has been squared, Notices Amer. Math. Soc. 36 (1989), pp. 1338–1343

    MathSciNet  MATH  Google Scholar 

  95. Garey M.R. and Johnson D.S., Computers and Intractability.A Guide to the Theory of NP-Completeness, Freeman, San Francisco, 1979

    MATH  Google Scholar 

  96. Gelfand I.M., Kapranov M.M. and Zelevinsky A.V., Newton Polytopes and the classical resultant and disciminant, Advances in Math. 84 (1990), pp. 237–254

    Article  MathSciNet  MATH  Google Scholar 

  97. Gerwien P., Zerschneidung jeder beliebigen Anzahl von gleichen gradlinigen Figuren in dieselben Stücke, J. reine angew. Math 10 (1833), pp. 228–234

    Article  MATH  Google Scholar 

  98. Gilbert E.N., A comparison of signaling alphabets. Bell System Tech. J. 31 (1952), pp. 504–522

    Google Scholar 

  99. Girard D. and Valentin P., Zonotopes and mixture management; In: New Methods in Optimization and their Industrial Uses, (ed. by J.P. Penot), ISNM87, Birkhäuser, Basel, 1989, pp. 57–71

    Google Scholar 

  100. Goldberg M., The isoperimetric problem for polyhedra, Tôhoku Math. J. 40 (1935), pp. 226–236

    Google Scholar 

  101. Goldfarb D. and Todd M.J., Linear programming; In: Handbooks in Operations Research and Management Science, Vol. 1, Optimization, (ed. by G.L. Nemhauser, A.G.H. Rinnooy Kan and M.J. Todd), North-Holland, Amsterdam, 1989, pp. 73–170

    Google Scholar 

  102. Grace D., Search for largest polyhedra, Math. Comput. 17 (1963), pp. 197–199

    Article  MATH  Google Scholar 

  103. Gritzmann P. and Hufnagel A., An algorithmic version of Minkowski’s reconstruction theorem, in preparation

    Google Scholar 

  104. Gritzmann P. and Klee V., On the 0-1 maximization of positive definite quadratic forms; In: Operations Research Proceedings, Springer, Berlin, 1989, pp. 222–227

    Google Scholar 

  105. Gritzmann P. and Klee V., Computational complexity of inner and outer j-radii of polytopes in finite dimensional normed spaces, Math. Programming 59 (1993), pp. 163–213

    Article  MathSciNet  MATH  Google Scholar 

  106. Gritzmann P. and Klee V., Mathematical programming and convex geometry In: Handbook of Convex Geometry, Vol. A, (ed. by P. Gruber and J. Wills), North-Holland, Amsterdam, 1993, pp. 627–674

    Google Scholar 

  107. Gritzmann P. and Klee V., On the complexity of some basic problems in computational convexity: I. Containment problems, Discrete Math., (1994) (to appear); Reprinted in: Trends in Discrete Mathematics, (ed. by W. Deuber, H.-J. Prömel und B. Voigt), Topics in Discrete Mathematics North-Holland, Amsterdam, 1994, to appear.

    Google Scholar 

  108. Gritzmann P. and Klee V., On the complexity of some basic problems in computational convexity: III. Probing and reconstruction, in preparation

    Google Scholar 

  109. Gritzmann P. and Klee V., On the complexity of some basic problems in computational convexity: IV. Some algebraic applications, in preparation

    Google Scholar 

  110. Gritzmann P., Klee V., and Larman D.G., Largest j-simplices in n-polytopes, preprint (1994)

    Google Scholar 

  111. Gritzmann P. and Sturmfels B., Minkowski addition of polytopes: computational complexity and applications to Gröbner bases, SIAM J. Discrete Math. 6 (1993), pp. 246–269

    Article  MathSciNet  MATH  Google Scholar 

  112. Gritzmann P. and Wills J.M., Lattice points; In: Handbook of Convex Geometry B, (ed. by P.M. Gruber and J.M. Wills), North-Holland, Amsterdam, 1993, pp. 765–798

    Google Scholar 

  113. Groemer H., On some mean values associated with a randomly selected simplex in a convex set, Pacific J. Math. 45 (1973), pp. 525–533

    Article  MathSciNet  MATH  Google Scholar 

  114. Grötschel M., Lovász L., and Schrijver A., The ellipsoid method and its consequences in combinatorial optimization, Combinatorica 1 (1981), pp. 169–197; Corr. 4 (1984), pp. 291-295

    MATH  Google Scholar 

  115. Grötschel M., Lovász L., and Schrijver A., Geometric Algorithms and Combinatorial Optimization, Springer, Berlin, 1988

    Book  MATH  Google Scholar 

  116. Grünbaum B., Convex Polytopes, Wiley-Interscience, London, 1967

    MATH  Google Scholar 

  117. Hadwiger H., Eulers Charakteristik und kombinatorische Geometrie, J. reine angew. Math. 194 (1955), pp. 101–110

    MathSciNet  MATH  Google Scholar 

  118. Hadwiger H., Vorlesungen überlnhalt, Oberfläche und Isoperimetrie, Springer, Berlin, 1957

    Chapter  Google Scholar 

  119. Hadwiger H., Zentralaffine Kennzeichnung des Jordanschen Inhalts, Elemente Math. 25 (1970), pp. 25–27

    MathSciNet  MATH  Google Scholar 

  120. Hadwiger H., Das Wills’sche Funktional, Monatshefte Math. 79 (1975), pp. 213–221

    Article  MathSciNet  MATH  Google Scholar 

  121. Hadwiger H. and Glur P., Zerlegungsgleichheit ebener Polygene, Elemente Math. 26 (1951), pp. 97–106

    MathSciNet  Google Scholar 

  122. Haiman M., A simple and relatively efficient triangulation of then-cube, Discrete Comput. Geom. 6 (1991), pp.287–289

    Article  MathSciNet  MATH  Google Scholar 

  123. Hardy G.H. and Wright E.M., An Introduction to the Theory of Numbers, Clarendon Press, 1968

    Google Scholar 

  124. Hilbert D., Mathematische Probleme, Nachr.Königl.Ges.Wiss.Göttingen, Math.-Phys. Kl. (1900), pp. 253–297; Bull. Amer. Math. Soc. 8 (1902), pp. 437-479.

    Google Scholar 

  125. Huber B. and Sturmfels B., A polyhedral method for solving sparse polynomial systems, Math. Comput., (1994) (to appear)

    Google Scholar 

  126. Hughes R.B., Minimum-cardinality triangulations of the d-cube ford=5 andd=6, Discrete Math. 118 (1993), pp.75–118

    Article  MathSciNet  MATH  Google Scholar 

  127. Hughes R.B. and Anderson M.R., Simplexity of the cube, manuscript, 1994

    Google Scholar 

  128. Jerrum M.R. and Vazirani U., A mildly exponential approximation algorithm for the permanent, manuscript, (1991)

    Google Scholar 

  129. Jessen B., The algebra of polyhedra and the Dehn-Sydler theorem, Math. Scand. 22 (1968), pp.241–256

    MathSciNet  MATH  Google Scholar 

  130. Jessen B., Zur Algebra der Polytope, Göttingen Nachr. Math. Phys. (1972), pp.47–53

    Google Scholar 

  131. John F., An inequality for convex bodies, Univ. of Kentucky, Research Club Bull. 8 (1942), pp. 8–11

    Google Scholar 

  132. John F., Extremum problems with inequalities as subsidiary condi-tions, In: Studies and Essays, Courant Anniversary Volume, Interscience, New York, 1948, pp. 187–204

    Google Scholar 

  133. Kaiser M., Deterministic algorithms that compute the volume of poly-topes, in preparation

    Google Scholar 

  134. Kannan R., Lovász L. and Simonovits M., Random walks, isotropic position, and volume algorithms, in preparation

    Google Scholar 

  135. Karmarkar N., Karp R., Lipton R., Lovász L., Luby M., A Monte-Carlo algorithm for estimating the permanent, SIAM J. Comput. 22 (1993), pp. 284–293

    Article  MathSciNet  MATH  Google Scholar 

  136. Karzanov A. and Khachiyan L.G. On the conductance of order Mar-cov chains, Rutgers Univ. Tech. Report (1990)

    Google Scholar 

  137. Kasimatis E.A. and Stein S.K., Equidissections of polygons, Discrete Math. 85 (1990), pp. 281–294

    Article  MathSciNet  MATH  Google Scholar 

  138. Kepler J., Nova Stereometria doliorum vinariorum, 1615; see: Johannes Kepler Gesammelte Werke, (ed. by M. Caspar), Beck, München, 1940

    Google Scholar 

  139. Khachiyan L.G., On the complexity of computing the volume of apolytope, Izvestia Akad. Nauk. SSSR, Engineering Cybernetics 3 (1988), pp.216–217

    Google Scholar 

  140. Khachiyan L.G., The problem of computing the volume of polytopesis #P-hard, Uspekhi Mat. Nauk. 44 (1989), pp.199–200

    MathSciNet  MATH  Google Scholar 

  141. Khachiyan L.G., Complexity of polytope volume computation: In: New Trends in Discrete and Computational Geometry, (ed by J. Pach), Springer, Berlin, 1993, pp. 91–101

    Chapter  Google Scholar 

  142. Khachiyan L.G. and Todd M.J., On the Complexity of approximating the maximal inscribed ellipsoid for a polytope, Math. Prog. 61 (1993), pp.137–159

    Article  MathSciNet  MATH  Google Scholar 

  143. Kingman J.F.C., Random secants of a convex body, J. Appl. Prob-ability 6 (1969), pp.660–672

    MathSciNet  MATH  Google Scholar 

  144. Kirszbraun M., Überdie zusammenziehenden und Lipschitzen Trans-formationen, Fund. Math. 22 (1934), pp.77–108

    Google Scholar 

  145. Klebaner F.C., Sudbury A. and Watterson G.A., The volumes of simplices, or find the penguin, J. Austral. Math. Soc. (ser. A) 47 (1989), pp.263–268

    Article  MathSciNet  MATH  Google Scholar 

  146. Klee V., Problem in barycentric coordinates, J. Appl. Physics 36 (1965), pp. 1854–1856

    Article  Google Scholar 

  147. Klee V. and Wagon S., Old and New Unsolved Problems in Plane Geometry and Number Theory, Math. Assoc. Amer., Washington, D.C., 1991

    MATH  Google Scholar 

  148. Kneser M., Einige Bemerkungen über das Minkowskische Flächen-mass, Archiv Math. 6 (1955), pp. 382–390

    Article  MathSciNet  MATH  Google Scholar 

  149. Knuth D., A permanent inequality, Amer. Math. Monthly (1981), pp. 731–740

    Google Scholar 

  150. Koshleva O. and Kreinovich V. Geombinatorics, computational complexity and saving environment: let’s start, Geombinatorics 3 (1994), pp. 90–99

    Google Scholar 

  151. Kozlov M.K., An approximate method of calculating the volume of aconvex polyhedron, USSR Comp. Math. Math. Phys. 22 (1982), pp. 227–233

    Article  MATH  Google Scholar 

  152. Kozlov M.K., Algorithms for volume computation, based on Laplacetransform, Technical Report; (1986), Computing Center USSR Acad. Sci.

    Google Scholar 

  153. Laczkovich M., Equidecomposability and discrepancy: a solution of Tarski’s circle-squaring problem, J. Reine Angew. Math. 404 (1990), pp. 77–117

    MathSciNet  MATH  Google Scholar 

  154. Landau H.J. and Slepian D., On the optimality of the regular simplex code, Bell System Tech. J. 45 (1966), pp. 1247–1272

    MathSciNet  MATH  Google Scholar 

  155. Lasserre J.B., An analytical expression and an algorithm for the volume of a convex polyhedron in R n, J. Opt. Th. Appl. 39 (1983), pp. 363–377

    Article  MathSciNet  MATH  Google Scholar 

  156. Lawrence J., Polytope volume computation, Math. Comput. 57 (1991), pp. 259–271

    Article  MathSciNet  MATH  Google Scholar 

  157. Lee Y.T. and Requicha A.A.G., Algorithms for computing the volume and other integral properties of solids. I. Known methods and open issues, Comm. ACM 25 (1982), pp. 635–641

    Article  MathSciNet  Google Scholar 

  158. Lee Y.T. and Requicha A.A.G., Algorithms for computing the volume and other integral properties of solids. II. A family of algorithms based on representation conversion and cellular approximation, Comm. ACM 25 (1982), pp. 642–650

    Article  MathSciNet  Google Scholar 

  159. Lin S.-Y. and Lin Y.-F., The n-dimensional Pythagorean theorem, Lin. Multilin. Algebra, 26 (1990), pp. 9–13

    Article  MATH  Google Scholar 

  160. Linial N., Hard enumeration problems in geometry and combina-torics, SIAM J. Algeb. Discr. Meth. 7 (1986), pp. 331–335

    Article  MathSciNet  MATH  Google Scholar 

  161. Lovász L., How to compute the volume?, Jahresbericht Deutsche Math. Verein. (1992), pp. 138–151

    Google Scholar 

  162. Lovász L., Random walks on graphs: A survey, In: Combinatorics:Paul Erdös is 80, Vol. II, (ed. by D. Miklos, V.T. Sós and T. Szönyi), Bolyai Society, Budapest, 1994, to appear.

    Google Scholar 

  163. Lovász L. and Simonivits M., The mixing rate of Markov chains, an isoperimetric inequality and computing the volume, Proc. 31st Ann. Symp. Found. Comput. Sci. (1990), pp. 364–455

    Google Scholar 

  164. Lovász L. and Simonivits M., Random walks in a convex body and an improved volume algorithm, Random Structures Alg. 4 (1993), pp. 359–412

    Article  MATH  Google Scholar 

  165. Maass W. and Turán G., On the complexity of learning from counterexamples, Proc. 30th Ann. IEEE Symp. Found. Comput. Sci. (1989), pp. 262–267

    Google Scholar 

  166. Martini H., Some results and problems around zonotopes, Coll. Math. Soc. J. Bolyai, 48, Intuitive Geometry, Siófok (1985), pp. 383–418

    MathSciNet  Google Scholar 

  167. Martini H., Cross-sectional measures In: Coll. Math. Soc. J. Bolyai, 63 (Intuitive Geometry, Szeged 1991, Norjh-Holland, Amsterdam, 1994, pp. ??-??

    Google Scholar 

  168. McKenna M. and Seidel R., Finding the optimal shadow of a convex polytope, Proc. Symp. Comp. Geom. (1985), pp. 24–28

    Google Scholar 

  169. McMullen P., The maximum number of faces of a convex polytope, Mathematika 17 (1970), pp. 179–184

    Article  MathSciNet  MATH  Google Scholar 

  170. McMullen P., Non-linear angle-sum relations for polyhedra, cones and polytopes, Math. Proc. Cambridge Philos. Soc. 78 (1975), pp. 247–261

    Article  MathSciNet  MATH  Google Scholar 

  171. McMullen P., Valuations and Euler-type relations of certain classes of convex polytopes, Proc. London Math. Soc. 35 (1977), pp. 113–135

    Article  MathSciNet  MATH  Google Scholar 

  172. McMullen P., Monotone translation invariant valuations on convex bodies, Arch. Math. 55 (1990), pp. 595–598

    Article  MathSciNet  MATH  Google Scholar 

  173. McMullen P., Valuations and dissections; In: Handbook of Convex Geometry B, (ed. by P.M. Gruber and J.M. Wills), North-Holland, Amsterdam, 1993, pp. 933–988

    Google Scholar 

  174. McMullen P. and Schneider R., Valuations on convex bodies; In: Convexity and its Applications, (ed. by P.M. Gruber and J.M. Wills), Birkhäuser, Basel, 1983, pp. 170–247

    Google Scholar 

  175. McMullen P. and Shephard G.C., Convex polytopes and the upper bound conjecture, Cambridge University Press, 1971

    Google Scholar 

  176. Mead D.G., Dissection of the hypercube into simplices, Proc. Amer. Math. Soc. 76 (1979), pp. 302–304

    Article  MathSciNet  MATH  Google Scholar 

  177. Miel G. and Monney R., On the condition number of Lagrangian numerical differentiation, Appl. Math. Comput. 16 (1985), pp. 241–252

    Article  MathSciNet  MATH  Google Scholar 

  178. Minkowski H., Allgemeine Lehrsätze über konvexe Polyeder, Nachr. Ges. Wiss. Göttingen (1897), pp. 198–219

    Google Scholar 

  179. Minkowski H., Volumen und Oberfläche, Math. Ann. 57 (1903), pp. 447–495

    Article  MathSciNet  MATH  Google Scholar 

  180. Minkowski H., Theorie der konvexen Körper, insbesondere Begründung ihres Oberflächenbegriffs; see: Gesammelte Abhandlun-gen, Vol. 2, Leipzig, Berlin, 1911

    Google Scholar 

  181. Monsky P., On dividing a square into triangles, Amer.Math. Monthly 77 (1970), pp. 161–164

    Article  MathSciNet  MATH  Google Scholar 

  182. Monsky P., A conjecture of Stein on plane dissections, Math. Z. 205 (1990), pp. 583–592

    Article  MathSciNet  MATH  Google Scholar 

  183. Montgomery H.L., Computing the volume of a zonotope, Amer. Math. Monthly 96 (1989), p. 431

    Article  Google Scholar 

  184. Moser W.O.J., Problems, problems, problems, Discrete Appl. Math. 31 (1991), pp. 201–225

    Article  MathSciNet  MATH  Google Scholar 

  185. Mulmuley K., Computational Geometry: An Introduction through Randomized Algorithms, Prentice Hall, New York, 1994

    Google Scholar 

  186. Nijenhuis A. and Wilf H.S., Combinatorial Algorithms, Academic Press, New York, 1978

    MATH  Google Scholar 

  187. Olmsted C., Two formulas for the general multivariate polynomial which interpolates a regular grid on a simplex, Math. Comput. 47 (1986), pp. 275–284

    Article  MathSciNet  MATH  Google Scholar 

  188. Ong M.E.G., Hierarchical basis preconditioned for second order elliptic problems in three dimensions, Ph.D. dissertation, Appl. Math. Dept., Univ. Wash., Seattle (1989)

    Google Scholar 

  189. Ong M.E.G., Uniform Refinement of a tetrahedron, SIAM J. Scientific Comput., to appear

    Google Scholar 

  190. O’Rourke J., Computational Geometry in C, Cambridge Univ. Press, 1994

    Google Scholar 

  191. O’Rourke J., Computational geometry column 22, SIGACT News 25 (1994), pp. 31–33

    Article  Google Scholar 

  192. Palmon O., The only convex body with extremal distance from the ball is the simplex, Israel J. Math. 80 (1992), pp. 337–349

    Article  MathSciNet  MATH  Google Scholar 

  193. Papadimitriou C.H. and Yannakakis M., On recognizing integer polyhedra, Combinatorica 10 (1990), pp. 107–109

    Article  MathSciNet  MATH  Google Scholar 

  194. Pedersen P. and Sturmfels B., Product formulas for resultants and Chow forms, Math. Z., 214 (1993), pp. 377–396

    Article  MathSciNet  MATH  Google Scholar 

  195. Pisier G., The volume of convex bodies and Banach space geometry, Cambridge University Press, 1989

    Google Scholar 

  196. Podkorytbv A.N., Summation of multiple Fourier series over poly-hedra (in Russian), Leningrad. Univ. Mat. Mekh. Astronom. 1 (1980), pp. 51–58

    Google Scholar 

  197. Preparata F.P. and Shamos M.I., Computational Geometry, Springer, New York etc., 1985

    Book  Google Scholar 

  198. Renegar J., The computational complexity and geometry of the first order theory of the reals. I. Introduction. Preliminaries. The geometry of semi-algebraic sets. The decision problem for the existential theory of the reals, J. Symbolic Comput. 13 (1992), pp. 255–299

    Article  MathSciNet  MATH  Google Scholar 

  199. Renegar J., The computational complexity and geometry of the first order theory of the reals. II. The general decision problem. Preliminaries for quantifier elimination, J. Symbolic Comput. 13 (1992), pp. 301–327

    Article  MathSciNet  MATH  Google Scholar 

  200. Renegar J., The computational complexity and geometry of the first order theory of the reals. III. Quantifier elimination, J. Symbolic Comput. 13 (1992), pp. 329–352

    Article  MathSciNet  MATH  Google Scholar 

  201. Rivlin T.J., Optimally stable Lagrangian numerical differentiation, SIAM J. numer. Anal. 12 (1975), pp. 712–725

    Article  MathSciNet  MATH  Google Scholar 

  202. Rivlin T.J., Chebyshev polynomials. From approximation theory to algebra and number theory. 2nd ed., John Wiley, New York, 1990

    MATH  Google Scholar 

  203. Ryser H., Combinatorial Mathematics, The Carus Mathematical Monographs, 14, Math. Assoc. Amer., Washington, D.C., 1963

    Google Scholar 

  204. Sah C.-H., Hilbert’s Third Problem: Scissors Congruence, Pitman, San Francisco, 1979

    MATH  Google Scholar 

  205. Sahni S., Computationally related problems, SIAM J. Comput. 3 (1974), pp. 262–279

    Article  MathSciNet  Google Scholar 

  206. Sallee F., A note on minimal triangulations of the n-cube, Discrete Appl. Math. 4 (1982), pp. 211–215

    Article  MathSciNet  MATH  Google Scholar 

  207. Salzer H.E., Some problems in optimally stable Lagrangian differentiation, Math. Comput. 28 (1974), pp. 1105–1115

    Article  MathSciNet  MATH  Google Scholar 

  208. Sangwine-Yager J.R., Mixed volumes, In: Handbook of Convex Geometry, (ed. by P.M. Gruber and J.M. Wills), North-Holland, Amsterdam, 1993, pp. 43–72

    Google Scholar 

  209. Schneider R., Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, Vol. 44, Cambridge University Press, 1993

    Google Scholar 

  210. Schneider R., Polyiopes and Brunn-Minkowski theory, THIS VOLUME, pp. 273–299

    Google Scholar 

  211. Schoen A.H., A defect-correction algorithm for minimizing the volume of a simple polyhedron which circumscribes a sphere. Research Report, Computer Science Dept., Southern Illinois Univ., Carbondale, 111. (1986)

    Google Scholar 

  212. Seidel R., Output-Size Sensitive Algorithms for Constructive Problems in Computational Geometry, Ph.D. Thesis, Department of Computer Science, Cornell University, Ithaca, New York, 1987

    Google Scholar 

  213. Seidel R., Small dimensional linear programming and convex hulls made easy, Discrete Comput. Geom. 6 (1991), pp. 423–434

    Article  MathSciNet  MATH  Google Scholar 

  214. Shephard G.C., The mean width of a convex polytope, J. London Math. Soc. 43 (1968), pp. 207–209

    Article  MathSciNet  MATH  Google Scholar 

  215. Shephard G.C., Combinatorial properties of associated zonotopes, Canad. J. Math. 26 (1974), pp. 302–321

    Article  MathSciNet  MATH  Google Scholar 

  216. Shoemaker D.P. and Huang T.C., A systematic method for calculating the volumes of polyhedra corresponding to Brillouin zones, Acta Cryst. 7 (1954), pp. 249–259

    Article  Google Scholar 

  217. Sinclair A. and Jerrum M., Approximate counting, uniform generation and rapidly mixing Markov chains, Inform. Comput. 82 (1989), pp. 93–133

    Article  MathSciNet  MATH  Google Scholar 

  218. Slepian D., The content of some extreme simplices, Pacific J. Math. 31 (1969), pp. 795–808

    Article  MathSciNet  MATH  Google Scholar 

  219. Sommerville D.M.Y., An Introduction to the Geometry of N Dimensions, Methuen, London, 1929

    MATH  Google Scholar 

  220. Speevak T., An efficient algorithm for obtaining the volume of a special kind of pyramid and application to convex polyhedra, Math. Comput. 46 (1986), pp. 531–536

    Article  MathSciNet  MATH  Google Scholar 

  221. Spirakis P.G., The volume of the union of many spheres and point location problems, Proc. Second Ann. Symp. Th. Comput. Sci.; Lecture Notes in Computer Science, No. 182, Springer, Berlin, 1985, pp. 328-338

    Google Scholar 

  222. Stanley R., Two combinatorial applications of the Aleksandrov-Fenchel inequalities, J. Comb. Th. A 17 (1981), pp. 56–65

    Article  MathSciNet  Google Scholar 

  223. Stanley R., Two order polyiopes, Discrete Comput. Geom.. 1 (1986), pp. 9–23

    Article  MathSciNet  MATH  Google Scholar 

  224. Stanley R., Enumerative Combinatorics, Vol. 1, Wadsworth-Brooks/Cole, Pacific Grove, California, 1986

    MATH  Google Scholar 

  225. Stanley R., A zonotope associated with graphical degree sequences; In: Applied geometry and discrete mathematics: The Victor Klee Festschrift, (ed. by P. Gritzmann, B. Sturmfels), Amer. Math. Soc. and Assoc. Comput. Mach., 1991, pp. 555–570

    Google Scholar 

  226. Steiner J., Über parallele Flächen, Jahresber. Preuss. Akad. Wiss., (1840), pp. 114–118; see: Gesammelte Werke, Vol. II, Reimer, Berlin, 1882, pp. 173-176

    Google Scholar 

  227. Stromberg K., The Banach-Tarski paradox, Amer. Math. Monthly 86 (1979), pp. 151–161

    Article  MathSciNet  MATH  Google Scholar 

  228. Stroud A.H., Approximate Calculation of Multiple Integrals, Prentice Hall, Englewood Cliffs, 1971

    MATH  Google Scholar 

  229. Struik D.J., A Source Book in Mathematics, 1200-1800, Harvard Univ. Press, Cambridge, Mass., 1969

    MATH  Google Scholar 

  230. Sturmfels B., On the decidability of Diophantine problems in combinatorial geometry, Bull. Amer. Math. Soc. 17 (1987), pp. 121–124

    Article  MathSciNet  MATH  Google Scholar 

  231. Sydler J.-P., Conditions necessaire et suffisante pour Inequivalence des polyèdres de l’espace euclidien ä troi dimensions, Comm. Math. Helv. 40 (1965), pp. 43–80

    Article  MathSciNet  MATH  Google Scholar 

  232. Swart G., Finding the convex hull facet by facet, J. Algorithms 6 (1985), pp. 17–48

    Article  MathSciNet  MATH  Google Scholar 

  233. Tanner R.M., Contributions to the simplex code conjecture, Tech. Report No. 6151-8, Information Systems Lab., Stanford Univ., (1970)

    Google Scholar 

  234. Tanner R.M., Some content maximizing properties of the regular simplex, Pacific J. Math. 52 (1974), pp. 611–616

    Article  MathSciNet  MATH  Google Scholar 

  235. Tarasov S.P., Khachiyan L.G. and Erlich I.I., The method of inscribed ellipsoids, Soviet Math. Doklady 37 (1988), pp. 226–230

    MATH  Google Scholar 

  236. Tarski A., A Decision Method for Elementary Algebra and Geometry, Univ. of California Press, Berkeley, 1961

    Google Scholar 

  237. Tchakaloff V., Formules de cubature mécaniques à coefficients non negatifs, Bull. Sci. Math. 81 (1957), pp. 123–134

    MathSciNet  MATH  Google Scholar 

  238. Thue Poulsen E., Problem 10, Math. Scand. 2 (1954), p. 346

    Google Scholar 

  239. Todd M.J., The Computation of Fixed Points and Applications, Lecture Notes in Economic and Mathematical Systems, no. 124, Springer, Berlin, 1976

    Google Scholar 

  240. Todd M.J. and Tuncel L., A new triangulation for simplicial algorithms, SIAM J. Discrete Math. 6 (1993), pp. 167–180

    Article  MathSciNet  MATH  Google Scholar 

  241. Valiant L.G., The complexity of computing the permanent, Theor. Comput. Sci 8 (1979), pp. 189–201

    Article  MathSciNet  MATH  Google Scholar 

  242. Van Leeuwen J. and Wood D., The measure problem for rectangular ranges in d-space, J. Algorithms 2 (1981), pp. 282–300

    Article  MathSciNet  MATH  Google Scholar 

  243. Verschelde J. and Cool R., Nonlinear reduction for solving deficient polynomial systems by continuation methods, Numer. Math. 63 (1992), pp. 263–282

    Article  MathSciNet  MATH  Google Scholar 

  244. Verschelde J., Verlinden P. and Cool R., Homotopies exploiting Newton polytopes for solving sparse polynomial systems, SIAM J. Numer. Anal. (1994), (to appear)

    Google Scholar 

  245. Von Hohenbalken B., Finding simplicial subdivisions of polytopes, Math. Prog. 21 (1981), pp. 233–234

    Article  MATH  Google Scholar 

  246. Wagon S., The Banach-Tarski Paradox, Cambridge Univ. Press, Cambridge, 1985

    Book  MATH  Google Scholar 

  247. Walkup D.W., A simplex with a large cross-section, Amer. Math. Monthly 75 (1968), pp. 34–36

    Article  MathSciNet  MATH  Google Scholar 

  248. Walkup D.W. and Wets R.J.B., Lifting projections of convex poly-hedra, Pacific J. Math 28 (1969), pp. 465–475

    Article  MathSciNet  MATH  Google Scholar 

  249. Weber C.L., Elements of Detection and Signal Design, McGraw-Hill, New York, 1968

    Google Scholar 

  250. Yao F., Computational geometry; In: Handbook of Theoretical Computer Science, A, (ed. by J. van Leeuwen), Elsevier, Amsterdam, 1990, pp. 345–390

    Google Scholar 

  251. Zamanskii L.Y. and Cherkasskii V.L., Determination of the number of integer points in polyhedra in R 3: polynomial algorithms, Dokl. Acad. Nauk Ukrain. USSR, Ser. A 4 (1983), pp. 13–15

    MathSciNet  Google Scholar 

  252. Zamanskii L.Y. and Cherkasskii V.L., Generalization of the Jacobi-Perron algorithm for determining the number of integer points in polyhedra, Dokl. Acad. Nauk USSR, Ser. A 10 (1985), pp. 10–13

    Google Scholar 

  253. Zhu X., Progress in evaluating permanents of matrices for HBT Event simulation, Manuscript, Nuclear Physics Lab, Univ. of Washington, Seattle (1993)

    Google Scholar 

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Authors and Affiliations

  1. Fb IV, Mathematik, Universität Trier, D-54286, Trier, Germany

    Peter Gritzmann

  2. Department of Mathematics, University of Washington, Seattle, WA, 98195, USA

    Victor Klee

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  1. Peter Gritzmann
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  2. Victor Klee
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  1. Department of Mathematics and Statistics, The University of Calgary, Calgary, Alberta, Canada

    T. Bisztriczky

  2. Department of Mathematics, University College, London, UK

    P. McMullen

  3. Mathematisches Institut, Universität Freiburg, Freiburg, Germany

    R. Schneider

  4. Department of Mathematics and Statistics, York University, North York, Ontario, Canada

    A. Ivić Weiss

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Gritzmann, P., Klee, V. (1994). On the Complexity of Some Basic Problems in Computational Convexity. In: Bisztriczky, T., McMullen, P., Schneider, R., Weiss, A.I. (eds) Polytopes: Abstract, Convex and Computational. NATO ASI Series, vol 440. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0924-6_17

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