Applied Geometr y and Discret e Mathematic s The Victo r Kle e Festschrif t This page intentionally left blank DIMACShttps://doi.org/10.1090/dimacs/004 Series in Discrete Mathematic s and Theoretical Computer Scienc e

Volume 4

Applied Geometr y and Discret e Mathematic s The Victo r Kle e Festschrif t

Peter Gritzman n Bernd Sturmfel s Editors

NSF Scienc e an d Technolog y Cente r in Discret e Mathematic s an d Theoretica l Compute r Scienc e A consortium o f Rutgers University , Princeto n University , AT&T Bell Labs , Bellcor e 198 0 Mathematics Subject Classification (198 5 Revision). Primary 05 , 15 , 28 , 46 , 51 , 52 , 57, 65 , 68 , 90 .

Library of Congress Cataloging-in-Publication Data Applie d geometr y an d discret e mathematics : Th e Victo r Kle e festschrift/Pete r Gritzmann , Bern d Sturmfels , editors . p. cm.—(DIMAC S serie s in discret e mathematic s an d theoretica l compute r science , ISS N 1052-1798 ; v. 4) Include s bibliographica l references . AMS : ISB N 0-8218-6593- 5 (acid-fre e paper ) ACM : ISB N 0-89791-385- X (acid-fre e paper ) 1. Mathematics . 2. Klee , Victor . I. Gritzmann , Peter , 1954 - . II. Sturmfels , Bernd , 1962- . III. Klee , Victor . IV . Series . QA7.A66 4 199 1 90-2693 4 510—dc2 0 CI P

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This DIMACS volume, The "Victor Kle e Festschrift," i s a collection o f research and survey papers that are related to the work of Victor Klee. Th e publication o f thi s boo k o n th e occasio n o f Professo r Klee' s 65t h birth - day mirrors the breadth o f his mathematical contributions . W e especially thank th e editors, Peter Gritzmann an d Bern d Sturmfels , fo r preparin g a volume tha t contain s article s o n suc h a variety o f subjects , an d tha t i s a suitable tribute to a leader in the field of discrete mathematics.

Daniel Gorenstein, Directo r Robert Tarjan , Co-Directo r Fred S . Roberts, Associate Directo r This page intentionally left blank Brief Contents

Preface i x Biography o f Victor Klee x i Bibliography o f Victor Klee xvi i Contents (i n alphabetical order by author) xxx i List o f Papers (b y subjects) xxx v Contributed Article s 1-60 8 This page intentionally left blank Preface

This volume comprises a collection of research articles dedicated to Victor Klee on the occasion of his 65th birthday in September 1990 . Al l papers are related t o Victo r Klee' s researc h work , and , i n vie w o f hi s broad interests , a wide range o f area s in mathematics an d it s applications ar e touched upo n here. Thes e areas include • Discrete and Computational Geometry , • Classical and Computational Convexity , • Convex Polytopes and their Relatives, • Combinatorics, Polyhedra l Combinatorics, an d Graph Theory , • Functional Analysis, • Mathematical Programmin g and Optimization, an d • Theoretical Computer Science . Victor Klee has made significant contribution s not only to all of the above fields, bu t als o t o mathematic s education , mathematica l method s i n eco - nomics an d th e decisio n sciences , application s o f discret e mathematic s i n the biological an d socia l sciences , an d informatio n linkag e between applie d mathematics an d industry . Rathe r tha n attemptin g t o summariz e o r com - ment o n Victo r Klee' s numerou s professiona l achievements , w e let hi s vita and bibliography spea k for themselves . Following th e spiri t o f Victo r Klee' s holisti c vie w o f mathematics , th e present collectio n i s no t divide d int o mathematica l subcategories , bu t th e articles appea r i n alphabetica l orde r b y first author . I n orde r t o facilitat e browsing through this volume and to give easy access to papers belonging to the same area, we include a list o f papers by subject area . We are indebted t o the Center fo r Discret e Mathematics an d Theoretica l Computer Science , in particular to its director Daniel Gorenstein, and to the American Mathematica l Societ y fo r thei r hel p i n arrangin g th e publicatio n of this volume. W e wish to thank th e referee s fo r thei r invaluabl e help an d the authors for their enthusiastic support throughout this project. But , above all, w e join al l contributor s i n thei r birthda y wishe s expressin g th e deepes t gratitude to Victor Kle e for al l that h e has given to us.

Peter Gritzmann an d Bern d Sturmfel s September 199 0

ix t^p

photograph by Lisette Klce — Victor Kle e — Biography of Victor Klee

Personal Born in Sa n Francisco, 192 5 Education Ph.D., University o f Virginia, 194 9 B.A., Pomona College , 194 5 Honorary Degree s D.Sc, Universite d e Liege, 198 4 D.Sc, Pomona College , 196 5 Awards Pomona Colleg e David Prescot t Barrow s Award fo r Distinguishe d Achievement , 1988 Reed Colleg e VoUum Award fo r Distinguishe d Accomplishmen t i n Scienc e and Technology, 198 2 Alexander von Humboldt Stiftun g Preistrager, 1980-198 1 Mathematical Associatio n o f Americ a C. B. Allendoerfer Award , 198 0 Annual Award fo r Distinguishe d Servic e to Mathematics, 197 7 L. R. Ford Award , 197 2 University o f Virgini a President's and Vistor's Research Prize, 195 2 Full-time Employmen t University o f Washingto n Professor o f Mathematics, 1957-presen t Associate Professor, 1954-195 7 Assistant Professor , 1953-195 4 xii BIOGRAPH Y O F VICTO R KLE E

Adjunct Professo r o f Computer Science , 1974-presen t Professor o r Adjunct Professo r o f Applied Mathematics, 1976-198 4 University o f Western Australi a Visiting Professor, 197 9 University o f Victori a Visiting Professor, 197 5 T. J. Watson Researc h Center , IB M Full-time Consultant, 197 2 University o f Colorad o Visiting Professor, 197 1 University o f California, Lo s Angeles Visiting Associate Professor, 1955-195 6 University o f Virgini a Assistant Professor , 1949-195 3 Instructor, 1947-194 8 Fellowships Senior Fellow, Institute fo r Mathematic s an d its Applications, Min - neapolis, 198 7 Mathematical Science s Research Institute , Berkeley , 1985-198 6 Guggenheim Fellow , University o f Erlangen-Nurnberg, 1980-198 1 Center fo r Advance d Stud y in the Behavioral Sciences , Stanford, 1975 - 1976 Sloan Foundation Fellow , University o f Copenhagen, 1959-196 0 National Scienc e Foundation Senio r Postdoctoral Fellow , Universit y of Copenhagen , 1958-195 9 Research Fello w of the Alfred P . Sloan Foundation, 1956-195 8 National Researc h Council , Institute fo r Advance d Study , 1951-195 2 A. E. C. Predoctoral, University o f Virginia, 1948-194 9 Du Pont Predoctoral , University o f Virginia, 1945-194 7 Part-time Consultan t W. H. Freeman an d Company, 1976-presen t Holt, Rinehart an d Winston, Inc. , 1966-197 6 E. I. du Pont d e Nemours, Inc., 1968-197 2 The RAND Corporation , 1966-197 0 Boeing Scientifi c Researc h Laboratories , 1963-196 9 Professional Societie s American Mathematica l Societ y Associate Secretary , 1955-195 8 BIOGRAPHY O F VICTO R KLE E xii i

Symposium o n Convexity , Chairman , Organizin g Committee, 196 1 Council, 1964-1966 , 1969-197 1 Executive Committee, 1969-197 0 American Association fo r the Advancement o f Scienc e Chairman o f Sectio n A , 197 5 Fellow, 1976-presen t Mathematical Associatio n o f Americ a Board o f Governors, 1967-197 8 First Vice-President , 1968-197 0 President-Elect, 1970-197 1 President 1971-197 3 Sigma Xi National Lecturer , 196 9 Society for Industria l an d Applied Mathematic s Council, 1966-196 8 Also member o f Association fo r Computin g Machinery, Internationa l Linear Algebra Society , Mathematical Programmin g Society , Operation s Research Society , and Phi Beta Kappa. Invited Lecture s International Congres s o f Mathematicians, Vancouver , 197 4 Eighth International Symposiu m o n Mathematical Programming , Stan - ford, 197 3 (Plenary Speaker ) Invited lecture s at various annual meetings o f American Mathemati - cal Society, Mathematical Associatio n o f America, Societ y fo r Industria l and Applied Mathematics, American Association fo r th e Advancemen t of Science , Canadian Mathematica l Congress , Deutsche Mathematike r Vereinigung, Deutsche Gesellschaf t fii r Mathematik , Okonomi e und Op- erations Researc h Invited hour addresses at national o r international conference s de - voted to the following subjects : Applications o f Combinatorics i n the Biological and Socia l Science s (1988) Applied Linea r Algebra (1988 ) Combinatorics an d Geometr y (1989 ) Discrete Optimization (1981 , 1980 , 1977 ) Discrete Geometr y (1981 , 1966 , 1962 ) Convexity (1980 , 1975 , 1965 , 1961 ) Information Linkag e between Applied Mathematics an d Industr y (1978) xiv BIOGRAPH Y O F VICTO R KLE E

Combinatorial Mathematic s (1990 , 1978 , 1969 , 1969 , 1968 , 1968, 1963) Computing in Algebra and Number Theor y (1975 ) Mathematical Programmin g (1988 , 1982 , 1973 , 1967 ) Set-Theoretic Topolog y (1973 , 1961 , 1955) Mathematical Method s o f Economic s (1972 ) Graph Theor y and its Applications (1983 , 1972 , 1969 ) Algorithmic and Applied Combinatorics (1986 , 1983 , 1971, 1969) Teaching of Geometry (1990 , 1988 , 1969 , 1967 ) Calculus of Variations and Control Theory (1968 ) Mathematics o f the Decision Science s (1967 ) Scientific Computin g (1964 ) Functional Analysis (1964 , 1960 , 1960 ) Polytopes and Conve x Set s (1990 ) Applied and Computational Convexit y (1990 ) Operations Researc h (1990 , 1989 , 1987 , 1983 , 1983 , 1980 ) Current Editorship s Discrete Mathematic s Discrete Applied Mathematic s Journal o f Combinatorial Theor y Linear Algebra and Application s Mathematics o f Operations Researc h Discrete and Computationa l Geometr y Ph.D. Student s 29 in Mathematic s 1 in Applied Mathematic s 1 in Computer Scienc e Some other activities o f the last six year s Institute fo r Mathematic s and its Application s Chairman o f Organizin g Committe e fo r Year-lon g Program i n Ap- plied Combinatoric s Coordinator, Progra m i n Discrete and Computationa l Geometr y Advisory Committee , Program i n Applied Linea r Algebr a Board o f Governor s Mathematical Science s Research Institut e Board o f Trustee s BIOGRAPHY O F VICTO R KLE E x v

Centre de Recherche de Mathematiques Appliquees, Universite d e Montreal Steering Committe e Cornell Universit y Advisory Panel, Center fo r Researc h i n Discrete Optimizatio n University o f Florid a Scientific Board , Cente r fo r Researc h i n Discrete Optimizatio n American Mathematical Societ y Nominating Committe e Centennial Fellowshi p Committe e Organizing Committee, Summe r Workshop on Mathematical Devel - opments Related to Linear Programmin g Invited speake r at specia l session s on differential equations , func - tional analysis, convex sets, combinatorics, discrete geometr y Mathematical Associatio n o f Americ a Committee o n the Annual Award for Distinguishe d Servic e Ad Hoc Committee o n Awards Mathematical Programmin g Societ y International Progra m Committe e Association fo r Computin g Machiner y Program Committe e fo r a symposium o n computational geometr y Operation Researc h Societ y o f Americ a Arranged specia l sessio n o f papers on mathematical aspect s o f lin- ear programmin g This page intentionally left blank Bibliography o f Victor Klee

1946 1. On the equation, (x) = 2m, Amer . Math . Monthl y 53, 327-328.

1947 2. On a conjecture of Carmichael, Bull. Amer . Math . Soc . 53 , 1183-1186 . 3. On completing a determinant, Amer . Math . Monthl y 54 , 96-97. 4. Some remarks on Euler's totient, Amer. Math . Monthl y 54 , 332.

1948 5. A generalization of Euler's <\>-function, Amer . Math . Monthl y 55 , 358-359. 6. The support property of a convex set in a linear normed space, Duke Math. J. 15 , 767-772.

1949 7. On a problem ofErdos, Amer . Math . Monthl y 56 , 21-22. 8. A note on FermaVs congruence, Amer. Math . Monthl y 56 , 626-628. 9. A characterization of convex sets, Amer. Math . Monthl y 56 , 247-249. 10. Dense convex sets, Duke Math. J . 16 , 351-354.

1950 11. Some characterization of reflexivity, Rev . Cienc . (Lima ) 52 , 15-23 . 12. Decomposition of an infinite-dimensional linear system into ubiquitous convex sets, Amer. Math . Monthl y 57 , 540-541 .

1951 13. Some characterizations of compactness, Amer . Math . Monthl y 58 , 389-393.

xvii XV111 BIBLIOGRAPHY O F VICTOR KLE E

14. On certain intersection properties of convex sets, Canad. J . Math. 3 , 272- 275. 15. Convex sets in linear spaces, Duke Math. J . 18 , 443-466. 16. Convex sets in linear spaces. II, Duke Math. J . 18 , 877-883.

1952 17. Invariant metrics in groups {solution of a problem ofBanach), Proc . Amer . Math. Soc . 3 , 484-487. 18. Convex functions and upper semi-continuous collections, (with R. D. An- derson), Duke Math. J . 19 , 349-357.

1953 19. The critical set of a convex body, Amer. J . Math. 75 , 178-188. 20. Convex sets in linear spaces. Ill, Duke Math. J . 20 , 105-112 . 21. Convex bodies and periodic homeomorphisms in Hilbert space, Trans . Amer. Math . Soc . 74 , 10-43 . 22. On a theorem ofBela Sz.-Nagy, Amer. Math . Monthl y 60 , 618-619.

1954 23. Invariant extension of linear functional, Pacifi c J . Math. 4 , 37-46. 24. Some remarks on continuous transformations, (wit h W . R . Utz) , Proc . Amer. Math . Soc . 5 , 182-184 . 25. A characterization of reflexivity by the lattice of closed subspaces, (with E. E. Floyd), Proc. Amer . Math . Soc . 5 , 655-661. 26. Common secants for plane convex sets, Proc . Amer . Math . Soc . 5 , 639-641.

1955 27. A note on extreme points, Amer. Math . Monthl y 62, 30-32. 28. Some topological properties of convex sets, Trans. Amer . Math . Soc . 78, 30-45. 29. Separation properties of convex cones, Proc. Amer . Math . Soc . 6 , 313- 318. 30. Boundedness and continuity of linear functional, Duk e Math. J . 22, 263- 270. 31. Some finite-dimensional affine topological spaces, Portugal. Math . 14 , 27-30. 32. On metric independence and linear independence, (wit h L . M . Blumen - thal), Proc. Amer . Math . Soc . 6 , 732-734. 33. Topological structure ofnormed linear spaces, Summary o f Lecture s an d Seminars, Summe r Institut e o n Se t Theoretic Topology , Madison , Wis - consin, Amer. Math . Soc , Providence, pp. 132-134 . BIBLIOGRAPHY O F VICTOR KLE E xix

1956 34. Solution of a problem ofE. M. Wright on convex functions, Amer . Math . Monthly 63, 106-107 . 35. A note on topological properties of normed linear spaces, Proc. Amer . Math. Soc . 7 , 673-67'4. 36. Strict Separation of Convex Sets, Proc. Amer . Math . Soc . 7 , 735-737. 37. The structure of semispaces, Math. Scand . 4 , 54-64. 38. An example in the theory of topological linear spaces, Arch. Math . 3 , 362-366. 39. Iteration of the 'Lin' operation for convex sets, Math . Scand . 4 , 231-238. 40. A note on certain function spaces, (with M . E . Rudin), Arch . Math . 7 , 469-470. 41. Fixed-point sets of periodic homeomorphisms of Hilbert space, Ann. o f Math. 64 , 393-395.

1957 42. Extremal structure of convex sets, Arch. Math . 8 , 234-240. 43. On a method of mapping due to Kadec and Bernstein, (wit h R. G. Long), Arch. Math . 8 , 280-285. 44. On a problem ofBanach, Colloq . Math . 5 , 78. 45. Homogeneity of infinite-dimensional parallelotopes, Ann. o f Math . 66 , 454-460.

1958 46. Extremal structure of convex sets. II , Math. Z . 69, 90-104. 47. On the Borelian and projective types of linear subspaces, Math. Scand . 6 , 189-199.

1959 48. Some characterizations of convex polyhedra, Acta Math. 102 , 79-107. 49. Some new results on smoothness and rotundity in normed linear spaces, Math. Ann . 139 , 51-63. 50. Continuous convex sets, (with D. Gale), Math. Scand . 7 , 379-391 .

1960 51. Polyhedral sections of convex bodies, Acta Math. 103 , 243-267. 52. An example related to the fixed-point property, Nieu w Arch . Wisk . 8 , 81-82. 53. Shrinkable neighborhoods in Hausdorff linear spaces, Math. Ann . 141 , 281-285. 54. Leray-Schauder theory without local convexity, Math . Ann . 141 , 286 - 296. (Correction s Math. Ann . 14 5 (1962), 464-465. ) xx BIBLIOGRAPH Y O F VICTOR KLE E

55. Asymptotes and projections of convex sets, Math. Scand . 8 , 356-362. 56. Mappings into normed linear spaces, Fund. Math . 49 , 25-34. 57. Circumspheres and inner products, Math. Scand . 8 , 363-370. 1961 58. Stability of the fixed-point property, Colloq. Math . 8 , 43-46. 59. Convexity of Chebyshev sets, Math. Ann . 142 , 292-304. 60. Topological equivalence of a Banach space with its unit cell, Bull. Amer . Math. Soc . 67 , 286-290. 61. Relative extreme points, Proceeding s o f the International Symposiu m o n Linear Spaces , Jerusalem, Israel , 1960 , pp. 286-290 . 62. A question of Katetov concerning the Hilbert parallelotope, Proc. Amer . Math. Soc . 12 , 900-903. 1962 63. A conjecture on weak compactness, Trans. Amer . Math . Soc . 104 , 394- 402. 64. Exotic topologies for linear spaces, Proceedings of the International Sym - posium o n Topology, Prague, Czechoslovakia, 1961 , pp. 238-249 . 1963 65. Topological structure of infinite-dimensional linear spaces: the classifica- tion problem, (specia l volum e devote d t o th e Internationa l Conferenc e on Functional Analysis , Warsaw, Poland, 1960) , Studia Math. 69-71 . 66. Barycentric calculus, Encyclopedia Brittanica , vol. 1 , p. 211. 67. The finite topology of a linear space, (with S . Kakutani), Arch. Math . 14 , 55-58. 68. Idempotency of the hull-formation H y, Z . Wahrscheinlichkeitstheorie 1 , 258-262. 69. The Euler characteristic in combinatorial geometry, Amer . Math . Monthly 70, 119-127 . 70. On a problem of Hirschfeld, Nieuw Arch. Wisk . 11 , 22-26. 71. Rearrangements of series of vectors, Math. Z . 81, 46-51. 72. On a question of Bishop and Phelps, Amer. J . Math. 85 , 95-98. 73. On a conjecture of Lindens trauss, Israel J. Math. 1 , 1-4 . 74. Helly's theorem and its relatives, (wit h L . Danze r an d B . Griinbaum) , in Convexit y (V . Klee, ed. ) Proc . Sympos . Pur e Math. , vol . 7 , Amer . Math. Soc , Providence, pp. 101-180 . 75. Topological classification of convex sets, (with H. Corson), Proc. Sympos . Pure Math. vol . 7 , Amer. Math . Soc , Providence, pp. 37-51 . 76. Infinite-dimensional intersection theorems, i n Convexit y (V . Klee , ed. ) Proc. Sympos . Pur e Math. , vol . 7 , Amer. Math . Soc , Providence , pp . 349-360. 77. The generation of affine hulls, Acta Sci . Math . (Szeged ) 24 , 60-81. BIBLIOGRAPHY O F VICTOR KLE E xx i

78. The generation of convex hulls, (wit h W . Bonnice) , Math . Ann . 152 , 1-29. 79. On a theorem ofDubins, J . Math. Anal . Appl . 7 , 425-427. 80. Convexity, Proc . Sympos . Pur e Math . vol . 7 , Amer . Math . Soc , Providence, 51 6 + x v pages, (Editor).

1964 81. Combinatorial geometry in the plane (wit h H. Hadwiger and H. Debrun- ner), Holt , Ne w York , 11 3 - h v pages , (Translato r an d autho r o f on e chapter). 82. Extreme points of convex sets without completeness of the scalar field, Mathematika 10 , 59-63. 83. Every simple closed curve in E 3 is unknotted in E 4 , (with R . H. Bing), J. London Math . Soc . 39 , 86-94. 84. On the angle between two lines in a Minkowski plane, (wit h P . Katz) , NieuwArch. Wisk . 12 , 102-105. 85. Connectedness in topological linear spaces, Israel J. Math. 2, 127-131 . 86. Some semicontinuity theorems for convex polytopes and cell-complexes, (with H . G. Egglesto n an d B . Griinbaum), Comment . Math . Helv . 39 , 165-188. 87. A 'string algorithm'for shortest paths in directed networks, Oper. Res . 12 , 428-432. 88. A combinatorial analogue ofPoincare's duality theorem, Canad. J . Math. 16, 517-531 . 89. Diameters of polyhedral graphs, Canad. J . Math. 16 , 602-614. 90. The number of vertices of a convex polytope, Canad . J . Math . 16 , 701-720. 91. A property of polyhedral graphs, J. Math. Mech . 13 , 1039-1042. 92. Utility functions and the 'lin' operation for convex sets, Israel J. Math. 2 , 191-197. 93. Two topological properties of topological linear spaces, (with C . Bessaga), Israel J. Math. 2 , 211-220.

1965 94. A theorem on convex kernels, Mathematika 12 , 89-93. 95. Summability in l(p {, p 2, ... ) spaces, Studia Math . 25 , 277-280. 96. Two examples in the theory of topological linear spaces, Studia Math. 25 , 385-390. 97. Problem in barycentric coordinates, J. Appl. Phys . 36 , 1854-1856 . 98. A class of linear programming problems requiring a large number of iter- ations, Numer. Math . 7 , 313-321 . 99. Heights of convex polytopes, J. Math. Anal . Appl . 11 , 176-190 . xxii BIBLIOGRAPH Y O F VICTOR KLEE

100. Paths on polyhedra, I, J. Soc . Indust . Appl . Math . 13 , 946-956. Translation: Russia n translation of Hadwiger-Debrunner (Izv . "Nauka, " Moscow, 1965 ) include s translation o f part o f m y added chapter . (Se e No. 81. )

1966 101. Every non-normable F-space is homeomorphic with its closed convex bodies, (with C. Bessaga), Math. Ann . 163 , 161-166 . 102. Convex polytopes and linear programming, Proceeding s of the IBM Sci- entific Computin g Symposium o n Combinatorial Problems , March 16 - 18, 1964 , IBM Data Processin g Division, pp. 123-158 . 103. Exposed points of convex sets, (with G. Choquet and H. Corson), Pacifi c J. Math. 16 , 33-43. 104. Paths on polyhedra. II , Pacific J. Math. 16 , 249-262. 105. A comparison of primal and dual methods for linear programming, Nu - mer. Math . 9 , 227-235; Reprinted: No . 10 1 in Contribution s t o functiona l analysis , Springer , New York.

1967 106. Remarks on nearest points in normed linear spaces, Proceedings o f th e Colloquium o n Convexity, Copenhagen , Denmar k 1965 , pp« 168-176 . 107. Problem size in linear programming, Proceeding s of the Colloquium o n Convexity, Copenhagen , Denmark, 1965 , pp. 177-184 . 108. The d-step conjecture for polyhedra of dimension d < 6 , (wit h D . Walkup), Acta Math. 117 , 53-78. 109. Lengths of snakes in boxes, (wit h L . Danzer) , J . Combin . Theor y 2, 258-265. 110. A method for constructing circuit codes, J. Assoc . Comput . Mach . 14 , 520-529. 111. Diameters of polytopes, Chapter 1 6 (pp. 341-355 ) o f Convex Polytopes, B. Griinbaum, Wiley , New York. 112. Long paths and circuits on polytopes, Chapter 1 7 (pp. 356-389 ) o f Con- vex Polytopes, B. Griinbaum, Wiley , New York. 113. Asymptotes of convex bodies, Math. Scand . 20 , 89-90. 114. Characterizations of a class of convex sets, (with C. Olech), Math. Scand . 20, 290-296. 115. Applications of geometry, Proceeding s o f the CUPM Geometry Confer - ence, Sant a Barbara , 1967 . Par t I : Convexit y an d Application s (L. Durst , ed.) , Mathematica l Associatio n o f America , Committe e o n the Undergraduate Progra m i n Mathematics, Berkeley , pp. 7-42 . BIBLIOGRAPHY O F VICTOR KLE E xxin

1968 116. Convex functions on convex poly topes, (with D . Gal e an d R . T . Rock - afellar), Proc . Amer . Math . Soc . 19 , 867-873. Ml, Behavior of linear forms on extreme points, Illinoi s J . Math . 12 , 254-263. 118. Maximal separation theorems for convex sets, Trans. Amer . Math . Soc . 134, 133-147 . 119. Facets and vertices of transportation polytopes, (wit h C . Witzgall), Lec- tures in Appl. Math . (G . Dantzig and A . Veinott, eds.) , vol. 11 , Amer. Math. Soc , Providence, pp. 257-282 . 120. Helly's theorem and its applications, (with L. Danzer and B. Griinbaum), (Expanded versio n o f No. 74 , translated int o Russia n b y S . Zalgaller), "Mir", Moscow , 16 0 pages. Reprinted: No . 9 8 in Lecture s i n Appl. Math. , vol . 11 , Amer. Math . Soc, Providence, pp. 65-76 .

1969 121. Convexity, Encyclopedi a Brittanica , pp. 436-437 . 122. Can a plane convex body have two equichordal points?, Amer . Math . Monthly 76 , 54-55. 123. Can nine tetrahedra form a neighboring family?, Amer . Math . Monthl y 76, 178-179 . 124. Is every polygonal plane region illuminable from some point?, Amer . Math. Monthl y 76, 180 . 125. What is the expected volume of a simplex whose vertices are chosen at random from a convex body?, Amer. Math . Monthl y 76 , 286-288. 126. Is there an n for which (x) = n has a unique solution?, Amer. Math . Monthly 76, 288-289. 127. Can the boundary of a d-dimensional convex body contain segments in all directions?, Amer. Math . Monthl y 76, 408-410. 128. Is a body spherical if its H A-measurements are constant?, Amer. Math . Monthly 76, 539-542. 129. Can all convex Borel sets be generated in a Borelian manner within the realm of convexity?, Amer. Math . Monthl y 76, 678-679. 130. Intersection theorems for positive sets, (wit h W . Hansen), Proc Amer . Math. Soc . 22 , 450-457. 131. What are the intersection graphs of arcs in a circle?, Amer . Math . Monthly 76, 810-813. 132. On a lemma of Fullerton and Braunschweiger, Math. Ann . 182 , 249- 250. 133. Invertibly positive linear operators on spaces of continuous functions, (with T. A. Brown and M. Juncosa), Math. Ann . 183 , 105-114 . XXIV BIBLIOGRAPHY O F VICTOR KLEE

134. Two renorming constructions related to a question of Anselone, Studi a Math. 33 , 231-242. 135. Separation and support properties of convex sets—a survey, Control The- ory an d th e Calculu s o f Variation s (A . Balakrishnan , ed.) , Academi c Press, New York, pp. 235-303 . 1970 136. What is the maximum length of a d-dimensional snake?, Amer. Math . Monthly 77 , 63-65. 137. Which isoperimetric ratios are bounded?, Amer . Math . Monthl y 77 , 288-289. 138. Must a compact endset have zero measure?, (wit h M . Martin) , Amer . Math. Monthl y 77, 616-618. 139. Convexite, Encyclopedia Universalis , 4, 982-985. 140. The use of circuit codes in analog-to-digital conversion, Graph Theor y and it s Application s (B . Harris, ed) , Academi c Press , Ne w York , pp . 121-131. 141. Shapes of the future—unsolved problems in geometry. Part I: two dimen- sions, 25-minut e colo r film an d 20-pag e viewer' s manual , Individua l Lecture Film Projec t o f the Mathematical Associatio n o f America . 1971 142. Semicontinuity of the face function of a convex set, (wit h M . Martin) , Comment. Math . Helv . 46 , 1-12 . 143. The use of research problems in high school geometry, Educationa l Stud - ies in Mathematics , vol . 3 , pp. 482-489 . 1971-7 2 (Reprinte d i n Th e Teaching o f Geometr y a t th e Pre-Colleg e Leve l (H.-G . Steiner , ed.) , Reidel, Dordrecht, pp . 206-213. ) 144. Shapes of the future, Amer . Sci . 59 , 84-91 (Reprinte d in Science Today (India) 5 , no . 12 , Jun e 1971 , 41-49, an d i n Th e Two-Yea r Colleg e Math. J . 2 , 14-27) . 145. The greedy algorithm for finitary and cofinitary matroids, Combinatorics (T. Motzkin , ed.) , Proc. Sympos . Pur e Math. , vol . 19 , Amer. Math . Soc, Providence , pp. 137-152 . 146. What is a convex set?, Amer. Math . Monthl y 78, 616-631. 147. Shapes of the future—unsolved problems in geometry. Part II: three di- mensions, 40-minute color film and 27-page viewer's manual, Individual Lecture Film Projec t o f the Mathematical Associatio n o f America . 148. Monthly research problems, 1969-1971 , (wit h R . Guy) , Amer . Math . Monthly 78, 1113-1122 . 1972 149. How good is the simplex algorithm?, (wit h G . Minty) , Inequalitie s II I (O. Shisha, ed), Academic Press, New York, pp. 159-175 . 150. Experimental designs by level reduction of the d-dimensional cuboctahe- dron, (wit h D. Doehlert), Discrete Math., 2 , 309-334. BIBLIOGRAPHY O F VICTOR KLE E XXV

151. Unions of increasing and intersections of decreasing sequences of convex sets, Israel J. Math., 12 , 70-78. 152. Which generalized prisms admit H-circuitsl, Grap h Theor y an d Appli- cations (Y . Alavi, D. R. Lick , and A . T. White, eds.), Springer, Berlin, pp. 173-178 . 153. On a question of Colin Clark concerning three properties of convex sets, Canad. Math . Bull. , 15 , 535-537. 1973 154. A remark on 'Some properties of ordered finite-dimensional spaces', Mathematical Model s in Economics (J . Los and M . W. Los, eds.), Pol- ish Scientifi c Publishers , Warsaw ; and North-Holland, Amsterdam , pp . 329-331. 1974 155. Polytope pairs and their relationship to linear programming, Act a Math. 133, 1-25 . 156. Shellings of spheres and poly topes, (wit h G . Danaraj) , Duk e Math . J . 41,443-451. 1975 157. Some proximate concepts in topology, (with A . Yandl), Sympos . Math . 16, 21-39. 158. Convex polyhedra and mathematical programming, Proceeding s o f th e 1974 International Congres s o f Mathematician s i n Vancouver , vo l 1 , pp. 485-490 . 159. Unique reducibility of subsets of commutative topological groups and semigroups, (with D. Gale), Math. Scand . 36 , 174-198 . 160. Spira's theorems on complete linear proofs of systems of linear inequali- ties, Mathematika 22 , 112-121 . 161. Ratio-sequences of chains in connected metric spaces, Th e Geometr y of Metri c an d Linea r Space s (L . M . Kelly , ed) , Springer , Berlin , pp. 134-146 . 162. A d-pseudomanifold with f 0 vertices has at least df 0- (d - \){d + 2) d-simplices, Houston J . Math. 1 , 81-86. 1976 163. Minimum graphs of specified diameter, connectivity and valence. I, (with H. Quaife), Math . Oper . Res . 1 , 28-31. 1977 164. A linearly compact convex set dense in every vector topology, Arch. Math . 28,80-81. 165. The connectedness game and the c-complexity of certain graphs, (wit h G. Danaraj), SIA M J. Appl. Math . 32 , 431-442. xxvi BIBLIOGRAPH Y O F VICTOR KLEE

166. When is a matrix sign stable?, (with C . Jeffries an d P . van de n Driess- che), Canad. J . Math. 29 , 315-326. 167. Can the measure of U"^ ? bj\ be computed in less than 0(n\ogn) steps?, Amer. Math . Monthl y 84, 284-285. 168. Classification and enumeration of minimum (d, 1 , 3)-graphs and min- imum (d, 2, 3)-graphs, (with H . Quaife) , J . Combin . Theor y B 23, 83-93. 169. Impressions of mathematical education in the People's Republic of China, Amer. Math . Monthl y 84, 509-517. 170. Pure and applied mathematics in the People's Republic of China (wit h E. Brown , G . Carrier , W . Feit , A . Fitzgerald , J . Keller , J . Kohn , C . Leban, S . MacLane, H. Pollak, and H. Wu) (S . MacLane and A. Fitzger- ald, eds.), National Academ y o f Sciences , Washington, D.C. , 11 6 + i x pages. 171. Linear algorithms for testing the sign stability of a matrix and for finding Z-maximum matchings in acyclic graphs, (with P. van de n Driessche) , Numer. Math . 28 , 273-285.

1978 172. Which spheres are shellable?, (with G. Danaraj), Algorithmi c Aspects of Combinatorics (B . Alspach e t al., eds), Ann. Discret e Math. 2 , 33-52. 173. A representation of two-dimensional pseudomanifolds and its use in the design of a linear-time shelling algorithm, (wit h G . Danaraj) , Algorith - mic Aspect s o f Combinatoric s (B . Alspach e t al. , eds) , Ann . Discret e Math. 2 , 53-63. 174. Adjoints of projective transformations and face-figures of convex poly- topes, Polyhedral Combinatoric s (M . Balinsk i an d A . Hoffman , eds.) , Mathematical Programmin g Studie s 8, 208-216.

1979 175. Use of Floyd's algorithm to find shortest restricted paths, (wit h D . Lar - man), Discret e Optimizatio n (P . Hammer e t al. , eds.) , Ann . Discret e Math. 4 , 237-249. 176. Combinatorial optimization: what is the state of the art?, Informatio n Linkage Betwee n Applie d Mathematic s an d Industr y (P . Wang , ed.) , Academic Press, New York, pp. 71-136 . 177. Some unsolved problems in plane geometry, Math . Mag . 52 , 131-145.

1980 178. On the complexity of d-dimensional Voronoi diagrams, Arch. Math . 34 , 75-80. 179. Another generalization of Caratheodory's theorem, Arch . Math . 34 , 560-562. BIBLIOGRAPHY O F VICTOR KLE E xxvi i

180. Classification and enumeration of minimum {d, 3 , 3)-graphsforodd d, J. Combin. Theor y B 28, 184-207 . 181. The diameter of almost all bipartite graphs, (wit h D . Larma n an d E . Wright), Studia Sci . Math . Hungar . 15 , 39-43. Updated version o f paper 17 6 above is in Math. Oper . Res . 5 , 1-26 . 1981 182. Qualitative matrices: strong sign-solvability and weak-satisfiability, (with R. Ladner) , Computer-Assiste d Analysi s an d Mode l Simplificatio n (H. Greenber g an d J . Maybee , eds.) , Academi c Press , Ne w York , pp. 293-320 . 183. Diameters of random graphs, (wit h D . Larman), Canad . J . Math . 33 , 618-640. 184. Dispersed Chebyshev sets and coverings by balls, Math . Ann . 257 , 251-260. 185. The proportion of labelled bipartite graphs which are connected, (with D. Larman an d E . Wright), J. London Math . Soc . 24 , 397-404. 1982 186. How many steps?, Convexit y an d Relate d Combinatoria l Geometr y (M. Breen and D. Kay, eds.), Marcel Dekker , New York, pp. 1-6 . 187. A note on convex cones and constraint qualifications in infinite-dimen- sional vector spaces, J. Optim. Theor y and Appl. 37 , 277-284. 1983 Chinese translation of paper 17 6 above appears in Applied Mathematics and Mathematics o f Computation (People' s Republic o f China ) 6 , 49- 65. 1984 188. Sign-solvability revisited, (wit h R . Ladner and R. Manber), Linear Alge- bra Appl. 59 , 131-157 . 189. Diameters of random bipartite graphs, (with B. Bollobas), Combinator - ica4, 7-19 . 1985 190. Finding the smallest triangles containing a given convex polygon, (wit h M. C. Laskowski), J. Algorithms 6 , 359-375. 1986 191. Inspheres and inner products, (wit h E . Maluta an d C . Zanco), Israe l J . Math. 55 , 1-14 . 192. Tiling with smooth and rotund tiles, (wit h E . Malut a an d C . Zanco) , Fund. Math . 126 , 269-290. XXV111 BIBLIOGRAPHY O F VICTOR KLE E

193. Limits of star shaped sets, (with G . Beer), Arch. Math . 46 , 241-249. 194. Facet centroids and volume minimization, Studi a Sci . Math . Hungar . 21, 143-147 . 195. Do infinite-dimensional Banach spaces admit nice tilings'], Studia Sci . Math. Hungar . 21,415-427 .

1987 196. Qualitative stability of linear systems, (wit h C . Jeffrie s an d P . van de n Driessche), Linear Algebra Appl. 87 , 1-48 . 197. Locally countable plump tilings are flat, (wit h C . Tricot), Math . Ann . 277, 315-325. 198. Recursive structure of S*-Matrices, and an 0(ra 2) algorithm for recog- nizing strong sign-solvability, Linea r Algebra Appl. 96 , 233-247. 199. The d-step conjecture and its relatives, (with P . Kleinschmidt) , Math . Oper. Res . 12 , 718-755.

1988 200. A qualitative analysis of x = Ax + b, (wit h T . Bon e an d C . Jeffries) , Discrete Appl. Math . 20 , 9-30.

1989 201. Sign-patterns and stability, Application s o f Combinatoric s an d Grap h Theory t o th e Biologica l an d Socia l Science s (F . Roberts , ed.) , IM A Volumes i n Mathematic s an d It s Applications, vol . 17 , Springer, Ne w York, pp. 203-219 . 202. On the 0- 1 maximization of positive definite quadratic forms, (wit h P . Gritzmann), Operation s Researc h Proceeding s 1988 , Springer , Berlin , pp. 222-227 . 203. Optimization of globally convex functions, (wit h T . C . H u an d D . G . Larman), SIA M J. Control Optim. 27 , 1026-1047 . 204. Edward James McShane, 1904-1989, (wit h W. Fleming), Notices Amer. Math. Soc . 36 , 828-830. 205. Minimum graphs of specified diameter, connectivity and valence. II , (with E . Engelhardt , K . Li , an d H . Quaife) , Discret e Math . 78 , 257-266.

1990 206. Geometry of the Gass-Saaty parametric cost LP algorithm, (wit h P. Klein- schmidt), Discrete Comput. Geom . 5 , 13-26 . 207. Computational complexity of norm-maximization, (wit h H. L. Bodlaen- der, P. Gritzmann, an d J. van Leeuwen) , Combinatorica 10 , 203-225. BIBLIOGRAPHY OF VICTOR KLE E xxi x

208. On the limited power of linear probes and other optimization oracles, (with P. Gritzmann and J. Westwater), Proceedings of the Sixth Annual Symposium o n Computationa l Geometry , Assoc , fo r Comput . Mach. , pp. 92-101 . 1991- 209. Good and bad radii of convex polygons, (with P. Gritzmann and L. Hab- sieger), SIAM J. Comput., 20 , 395-405. 210. Convex poly topes and related complexes, (wit h P. Kleinschmidt), Hand - book of Combinatorics (R . Graham, M. Grotschel and L. Lovasz, eds.), North-Holland, Amsterdam , accepte d fo r publication , 1991 . 211. Convex geometry in undergraduate instruction, (t o appear). 212. Old and new unsolved problems in plane geometry and number theory, (with S . Wagon), Mathematical Associatio n o f America, (t o appear). 213. Inner and outer j-radii of convex bodies in finite-dimensional normed spaces, (with P. Gritzmann), Discrete Comput. Geom. , (t o appear). 214. Computational complexity of inner and outer j-radii of convex poly topes, (with P. Gritzmann), Math. Prog. , (t o appear). 215. Uniform properties of collections of convex bodies, (with E . Maluta an d C. Zanco) (t o appear). E-mail address: [email protected] U This page intentionally left blank Contents

(in alphabetica l orde r b y author )

A Dual Fores t Algorith m fo r th e Assignment Proble m HANS ACHATZ , PETE R KLEINSCHMIDT , AN D KONSTANTINOS PAPARRIZO S 1 Self-duality Group s an d Rank s o f Self-dualitie s JONATHAN ASHLEY , BRANK O GRUNBAUM , G . C . SHEPHARD , AN D WALTER STROMQUIS T 1 1 Do Projection s G o t o Infinity ? IMRE BARANY , JACO B E . GOODMAN , AN D RICHAR D POLLAC K 5 1

The Minima l Projectiv e Plan e Polyhedra l Map s D. W . BARNETT E 6 3 Packing Euclidea n Spac e with Congruen t Cylinder s an d with Congruen t Ellipsoid s A. BEZDE K AN D W . KUPERBER G 7 1

Extended Euler-Poincar e Relation s fo r Cel l Complexe s ANDERS BJORNE R AN D GI L KALA I 8 1 Computing th e Conve x Hul l i n the Euclidea n Plan e in Linea r Expecte d Tim e KARL HEIN Z BORGWARDT , NORBER T GAFFKE , MICHAEL JUNGER , AN D GERHAR D REINEL T 9 1

Measures o f F-Star s i n Finitel y Starlik e Set s MARILYN BREE N 10 9 On Sign-Nonsingula r Matrice s an d th e Conversio n o f the Permanen t int o the Determinan t RICHARD A . BRUALD I AN D BRYA N L . SHADE R 11 7

Recognizing Propertie s o f Periodi c Graph s EDITH COHE N AN D NIMRO D MEGIDD O 13 5

XXXI xxxii CONTENT S

On Generi c Globa l Rigidit y ROBERT CONNELL Y 14 7

Some Regula r Map s an d Thei r Polyhedra l Realization s H. S . M . COXETE R AN D G . C . SHEPHAR D 15 7

Volumes o f a Random Polytop e i n a Convex Se t L. DALL A AN D D . G . LARMA N 17 5

Bodies o f Constan t Widt h i n Riemannian Manifold s an d Spaces o f Constan t Curvatur e B. V . DEKSTE R 18 1

Uniquely Remota l Hull s DUANE DETEMPLE , JAC K ROBERTSON , AN D GRAHA M WOO D 19 3

The Symmetrie s o f the Cu t Polytop e an d o f Som e Relative s M. DEZA , V . P . GRISHUKHIN , AN D M . LAUREN T 20 5

Complete Description s o f Smal l Multicu t Polytope s M. DEZA , M . GROTSCHEL , AN D M . LAUREN T 22 1

A Hyperplane Incidenc e Proble m wit h Application s t o Counting Distance s HERBERT EDELSBRUNNE R AN D MICH A SHARI R 25 3

Gaps i n Differenc e Sets , and th e Grap h o f Nearly Equa l Distance s PAUL ERDOS , ENDR E MAKAI , JANO S PACH , AN D JOE L SPENCE R 26 5

Remarks o n 5-Neighbo r Packing s an d Covering s with Circle s G. FEJE S TOT H AN D L . FEJE S TOT H 27 5

Symmetric Solution s t o Isoperimetric Problem s fo r Polytope s P. FILLIMA N 28 9

A Global Newton Metho d A. A . GOLDSTEI N 30 1

Volume Approximation o f Conve x Bodie s by Circumscribe d Polytope s PETER M . GRUBE R 30 9

Points Set s with Smal l Integra l Distance s HEIKO HARBORT H AN D LOTHA R PIEPMEYE R 31 9

Convex Minimizer s o f Variational Problem s ERHARD HEI L 32 5

Flattening a Rooted Tre e PAUL HILFINGER , EUGEN E L . LAWLER , AN D GUNTE R ROT E 33 5 CONTENTS xxxii i

The Geometri c Complementarit y Proble m an d Transcendin g Stationarity i n Globa l Optimizatio n REINER HORS T AN D HOAN G TU Y 34 1 Every Tre e i s Graceful (Bu t Som e ar e More Gracefu l tha n Others ) T. C . H u AN D A . B . KAHN G 35 5 Qualitative Analysi s o f Schu r Complement s CHARLES R . JOHNSO N AN D JOH N MAYBE E 35 9 Centers an d Invarian t Point s o f Conve x Bodie s M. J . KAISER , T . L . MORIN , AN D T . B . TRAFALI S 36 7 The Diameter o f Graph s o f Conve x Polytope s an d /-Vecto r Theor y GIL KALA I 38 7 Multiply Perspectiv e Simplices , Desmi c Triad s and th e Edelstei n Theorem s L.M.KELLY 41 3 Submanifolds o f the Cub e W. KUHNE L AN D CH . SCHUL Z 42 3 Finite Union s o f Close d Subgroup s o f the ^-Dimensiona l Toru s JIM LAWRENC E 43 3 Regular Triangulation s o f Conve x Polytope s CARL W . LE E 44 3 On the Number o f Antipodal o r Strictl y Antipodal Pair s of Point s i n Finit e Subset s o f R d E. MAKAI , JR . AN D H . MARTIN I 45 7 Multi-Order Convexit y JUAN-ENRIQUE MARTINEZ-LEGA Z AN D IVA N SINGE R 47 1 Almost Orthogona l Line s i n E d MOSHE ROSENFEL D 48 9 Chiral Polytope s EGON SCHULT E AN D ASI A IVI C WEIS S 49 3 Exact Uppe r Bound s fo r th e Number o f Face s in ^-Dimensiona l Voronoi Diagram s RAIMUND SEIDE L 51 7 Stretchability o f Pseudoline s i s NP-Har d PETER W.SHO R 53 1 A Zonotope Associate d wit h Graphica l Degre e Sequence s RICHARD P . STANLE Y 55 5 xxxiv CONTENT S

Geometry o f Space s of Homogeneous Polynomial s on Banach Lattice s K. SUNDARESA N 57 1 The Combinatorics o f Bivariate Spline s WALTER WHITELE Y 58 7 List of Papers

(by subjects )

Algebraic Combinatoric s Geometr y

Extended Euler-Poincar e Relation s fo r Cel l Complexe s A. BJORNE R AN D G . KALA I 8 1

The Diamete r o f Graph s o f Conve x Polytope s an d /-Vecto r Theor y G. KALA I 38 7

Chiral Polytope s E. SCHULT E AN D A . Ivic WEIS S 49 3

A Zonotope Associate d wit h Graphica l Degre e Sequence s R. P . STANLE Y 55 5

Combinatorial Geometr y

Self-duality Group s an d Rank s o f Self-dualitie s J. ASHLEY , B . GRUNBAUM , G . C . SHEPHARD , AND W . STROMQUIS T 1 1

The Minima l Projectiv e Plan e Polyhedra l Map s D. W . BARNETT E 6 3

Measures o f ^-stars i n Finitel y Starlik e Set s M. BREE N 10 9

Some Regula r Map s an d Thei r Polyhedra l Realization s H. S . M . COXETE R AN D G . C . SHEPHAR D 15 7

A Hyperplane Incidenc e Proble m wit h Application s t o Counting Distance s H. EDELSBRUNNE R AN D M . SHARI R 25 3

Point Set s with Smal l Integra l Distance s H. HARBORT H AN D L . PIEPMEYE R 31 9

XXXV xxxvi PAPER S BY SUBJECT

Multiply Perspectiv e Simplices , Desmic Triad s and th e Edelstei n Theorem s L.M.KELLY 41 3

On th e Number o f Antipodal o r Strictl y Antipoda l Pair s o f Point s i n Finite Subset s i n R d E. MAKAI , JR . AN D H . MARTIN I 45 7

Almost Orthogona l Line s i n E d M. ROSENFEL D 48 9

Combinatorial Optimizatio n A Dual Fores t Algorith m fo r th e Assignment Proble m H. ACHATZ , P . KLEINSCHMIDT , AN D K . PAPARRIZO S 1

The Symmetrie s o f the Cu t Polytop e an d o f Som e Relative s M. DEZA , V . P . GRISHUKHIN , AN D M . LAUREN T 20 5

Complete Description s o f Smal l Multicut Polytope s M. DEZA , M . GROTSCHEL , AN D M . LAUREN T 22 1

Combinatorial Topolog y

Extended Euler-Poincar e Relation s fo r Cel l Complexe s A. BJORNE R AN D G . KALA I 8 1

Submanifolds o f th e Cub e W. KUHNE L AN D CH . SCHUL Z 42 3

Finite Union s o f Close d Subgroup s o f the ^-Dimensiona l Toru s J. LAWRENC E 43 3

Computational Geometr y

Computing th e Conve x Hul l i n the Euclidea n Plan e i n Linear Expecte d Tim e K. H . BORGWARDT , N . GAFFKE , M . JUNGER , AN D G . REINEL T 9 1

A Hyperplane Incidenc e Proble m wit h Application s t o Counting Distance s H. EDELSBRUNNE R AN D M . SHARI R 25 3

Exact Upper Bound s fo r th e Number o f Face s i n ^-Dimensiona l Voronoi Diagram s R. SEIDE L 51 7

Stretchability o f Pseudoline s I s NP-Har d P. W . SHO R 53 1 PAPERS BY SUBJECT xxxvi i

Computer Scienc e Recognizing Properties o f Periodic Graph s E. COHE N AN D N. MEGIDD O 13 5 Flattening a Rooted Tre e P. HlLFINGER , E . L . LAWLER , AN D G . ROT E 33 5

Convex Geometry Volumes o f a Random Polytop e i n a Convex Se t L. DALL A AN D D . G . LARMA N 17 5 Uniquely Remota l Hull s D. DETEMPLE , J . M . ROBERTSON , AN D G . WOO D 19 3 Symmetric Solutions to Isoperimetric Problems fo r Polytope s P. FILLIMA N 28 9 Volume Approximation o f Conve x Bodies by Circumscribed Polytope s P. M . GRUBE R 30 9 Convex Minimizers o f Variational Problem s E. HEI L 32 5 Centers and Invariant Point s of Conve x Bodie s M. J . KAISER , T . L . MORIN , AN D T . TRAFALI S 36 7 On the Number o f Antipodal or Strictl y Antipodal Pair s o f Points in Finite Subset s in R d E. MAKAI , JR. AN D H . MARTIN I 45 7

Convexity in General Spaces, Generalized Convexit y Bodies o f Constant Widt h i n Riemannian Manifold s and Space s of Constant Curvatur e B. V . DEKSTE R 18 1 Multi-Order Convexit y J-E MARTINEZ-LEGA Z AN D I . SINGE R 47 1 Geometry o f Space s of Homogeneous Polynomial s o n Banach Lattice s K. SUNDARESA N 57 1

Discrete Geometr y Do Projections G o to Infinity ? I. BARANY , J . E . GOODMAN , AN D R . POLLAC K 5 1 xxxviii PAPER S BY SUBJECT

Packing Euclidean Spac e with Congruent Cylinder s an d with Congruent Ellipsoid s A. BEZDE K AN D W . KUPERBER G 7 1 On Generic Global Rigidit y R. CONNELL Y 14 7 Gaps i n Difference Sets , and the Graph o f Nearly Equa l Distance s P. ERDOS , E . MAKAI , J . PACH , AN D J . SPENCE R 26 5 Remarks on 5-Neighbo r Packing s and Covering s G. FEJE S TOT H AN D L . FEJE S TOT H 27 5

Functional Analysis Geometry o f Space s of Homogeneous Polynomials o n Banach Lattice s K. SUNDARESA N 57 1

Graph Theory Recognizing Properties o f Periodic Graph s E. COHE N AN D N . MEGIDD O 13 5 Gaps in Differenc e Sets , and the Graph o f Nearly Equa l Distance s P. ERDOS , E . MAKAI , J . PACH , AN D J . SPENCE R 26 5 Every Tree is Graceful (Bu t Som e Are More Graceful tha n Others ) T. C . H u AN D A . B . KAHN G 35 5 A Zonotope Associate d with Graphica l Degre e Sequence s R. P . STANLE Y 55 5

Mathematical Programmin g A Dual Forest Algorith m fo r the Assignment Proble m H. ACHATZ , P . KLEINSCHMIDT , AN D K . PAPARRIZO S 1 The Symmetrie s o f the Cut Polytope and o f Som e Relative s M. DEZA , V . P . GRISHUKHIN , AN D M . LAUREN T 20 5 Complete Description s o f Smal l Multicut Polytope s M. DEZA , M . GROTSCHEL , AN D M . LAUREN T 22 1 A Global Newton Metho d A. A . GOLDSTEI N 30 1 The Geometric Complementarity Proble m an d Transcendin g Stationarity i n Global Optimizatio n R. HORS T AN D H . TU Y 34 1 PAPERS B Y SUBJECT xxxi x

Centers and Invariant Point s o f Conve x Bodie s M. J . KAISER , T . L . MORIN , AN D T . TRAFALI S 36 7

Matrix Theory On Sign-Nonsingula r Matrice s and the Conversion o f the Permanent int o the Determinan t R. A . BRUALD I AN D B . L . SHADE R 11 7 Qualitative Analysi s of Schu r Complement s C. R . JOHNSO N AN D J . MAYBE E 35 9

Nonlinear an d Global Optimizatio n A Global Newton Metho d A. A . GOLDSTEI N 30 1 The Geometric Complementarity Proble m an d Transcendin g Stationarity i n Global Optimizatio n R. HORS T AN D H . TU Y 34 1

Packing an d Covering Packing Euclidean Spac e with Congruent Cylinder s an d with Congruen t Ellipsoid s A. BEZDE K AN D W . KUPERBER G 7 1 Remarks o n 5-Neighbo r Packing s and Covering s G. FEJE S TOT H AN D L . FEJE S TOT H 27 5

Polyhedra Self-duality Group s and Rank s o f Self-dualitie s J. ASHLEY , B . GRUNBAUM , G . C . SHEPHARD , AND W . STROMQUIS T 1 1 The Minimal Projectiv e Plan e Polyhedral Map s D. W . BARNETT E 6 3 Some Regular Maps and Their Polyhedral Realization s H. S . M . COXETE R AN D G . C . SHEPHAR D 15 7 Symmetric Solution s to Isoperimetric Problems fo r Polytope s P. FILLIMA N 28 9 The Diameter o f Graphs o f Conve x Polytopes and /-Vector Theor y G. KALA I 38 7 Finite Unions o f Close d Subgroup s o f the A2-Dimensiona l Toru s J. LAWRENC E 43 3 xl PAPERS BY SUBJECT

Regular Triangulations o f Conve x Polytope s C. W. LE E 44 3 Chiral Polytope s E. SCHULT E AN D A . Ivic WEIS S 49 3 Exact Upper Bounds for th e Number o f Faces in ^-Dimensiona l Voronoi Diagram s R. SEIDE L 51 7

Rigidity Theory and Splines On Generic Global Rigidit y R. CONNELL Y 14 7 The Combinatorics o f Bivariate Spline s W. WHITELE Y 58 7

Stochastic Geometry Computing the Conve x Hull in the Euclidean Plan e in Linear Expecte d Tim e K. H . BORGWARDT , N . GAFFKE , M . JUNGER , AN D G . REINEL T 9 1 Volumes o f a Random Polytop e in a Convex Se t L. DALL A AN D D . G . LARMA N 17 5

Theory of Variations Convex Minimizers o f Variational Problem s E. HEI L 32 5