Bulletin of the American Mathematical Society

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Book Information:

Author: Kazuo Murota
Title: Discrete convex analysis
Additional book information: SIAM Monographs on Discrete Mathematics and Applications, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003, xxii+389 pp., ISBN 0-89871-540-7, $111.00$

Imre Bárány and Jiří Matoušek, A fractional Helly theorem for convex lattice sets, Adv. Math. 174 (2003), no. 2, 227–235. MR 1963693, DOI 10.1016/S0001-8708(02)00037-3

Michel Brion, Points entiers dans les polyèdres convexes, Ann. Sci. École Norm. Sup. (4) 21 (1988), no. 4, 653–663 (French). MR 982338

J. W. S. Cassels, An introduction to the geometry of numbers, Classics in Mathematics, Springer-Verlag, Berlin, 1997. Corrected reprint of the 1971 edition. MR 1434478

[DK03]

V.I. Danilov and A. Koshevoy, Discrete convexity and unimodularity. I, preprint arXiv:math.CO/0304291, 2003.

Jean-Paul Doignon, Convexity in cristallographical lattices, J. Geom. 3 (1973), 71–85. MR 387090, DOI 10.1007/BF01949705

Selected translations, Birkhäuser Verlag, Basel, 1992. Selecta Math. Soviet. 11 (1992), no. 2. MR 1166622

Martin Grötschel, László Lovász, and Alexander Schrijver, Geometric algorithms and combinatorial optimization, 2nd ed., Algorithms and Combinatorics, vol. 2, Springer-Verlag, Berlin, 1993. MR 1261419, DOI 10.1007/978-3-642-78240-4

Jim Lawrence, Rational-function-valued valuations on polyhedra, Discrete and computational geometry (New Brunswick, NJ, 1989/1990) DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 6, Amer. Math. Soc., Providence, RI, 1991, pp. 199–208. MR 1143297

Jiří Matoušek, Lectures on discrete geometry, Graduate Texts in Mathematics, vol. 212, Springer-Verlag, New York, 2002. MR 1899299, DOI 10.1007/978-1-4613-0039-7

Christos H. Papadimitriou, The adjacency relation on the traveling salesman polytope is NP-complete, Math. Programming 14 (1978), no. 3, 312–324. MR 503750, DOI 10.1007/BF01588973

A. V. Pukhlikov and A. G. Khovanskiĭ, The Riemann-Roch theorem for integrals and sums of quasipolynomials on virtual polytopes, Algebra i Analiz 4 (1992), no. 4, 188–216 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 4 (1993), no. 4, 789–812. MR 1190788

Herbert E. Scarf, Test sets for integer programs, Math. Programming 79 (1997), no. 1-3, Ser. B, 355–368. Lectures on mathematical programming (ismp97) (Lausanne, 1997). MR 1464774, DOI 10.1016/S0025-5610(97)00058-0

Alexander Schrijver, Combinatorial optimization. Polyhedra and efficiency. Vol. A, Algorithms and Combinatorics, vol. 24, Springer-Verlag, Berlin, 2003. Paths, flows, matchings; Chapters 1–38. MR 1956924

[BM03]

I. Bárány and J. Matousek, A fractional Helly theorem for convex lattice sets, Adv. Math. 174 (2003), 227-235. MR 1963693

[Br88]

M. Brion, Points entiers dans les polyèdres convexes (French), Ann. Sci. École Norm. Sup. (4) 21 (1988), 653-663. MR 0982338

[C97]

J.W.S. Cassels, An Introduction to the Geometry of Numbers, Corrected reprint of the 1971 edition. Classics in Mathematics, Springer-Verlag, Berlin, 1997. MR 1434478

[DK03]

V.I. Danilov and A. Koshevoy, Discrete convexity and unimodularity. I, preprint arXiv:math.CO/0304291, 2003.

[Do73]

J.-P. Doignon, Convexity in cristallographical lattices, J. Geometry 3 (1973), 71-85. MR 0387090

[G74]

M.B. Gromova, The Birkhoff-von Neumann theorem for polystochastic matrices (Russian), Operations Research and Statistical Simulation, No. 2, Izdat. Leningrad. Univ., Leningrad, 1974, pp. 3-15; Selected translations. Selecta Math. Soviet. 11 (1992), no. 2, 145-158. MR 1166622, MR 52:448

[GLS93]

M. Grötschel, L. Lovász, and A. Schrijver, Geometric Algorithms and Combinatorial Optimization (Second edition), Algorithms and Combinatorics, 2, Springer-Verlag, Berlin, 1993. MR 1261419

[L91]

J. Lawrence, Rational-function-valued valuations on polyhedra, Discrete and computational geometry (New Brunswick, NJ, 1989/1990), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 6, Amer. Math. Soc., Providence, RI, 1991, pp. 199-208. MR 1143297

[M02]

J. Matousek, Lectures on Discrete Geometry, Graduate Texts in Mathematics, 212, Springer-Verlag, New York, 2002. MR 1899299

[P78]

C.H. Papadimitriou, The adjacency relation on the traveling salesman polytope is NP-complete, Math. Programming 14 (1978), 312-324. MR 0503750

[PK92]

A.V. Pukhlikov and A.G. Khovanskii, The Riemann-Roch theorem for integrals and sums of quasipolynomials on virtual polytopes (Russian), Algebra i Analiz 4 (1992), 188-216; translation in St. Petersburg Math. J. 4 (1993), no. 4, 789-812. MR 1190788

[S97]

H.E. Scarf, Test sets for integer programs. Lectures on mathematical programming (ismp97) (Lausanne, 1997), Math. Programming, Ser. B 79 (1997), 355-368. MR 1464774

[Sc03]

A. Schrijver, Combinatorial Optimization. Polyhedra and efficiency. Vols. A-C, Algorithms and Combinatorics, 24, A-C, Springer-Verlag, Berlin, 2003. MR 1956924, MR 2004b:90004b, MR 1956924