applied geometry and discrete mathematics · 2019-02-12 · fields, but also to mathematics...

Applied Geometr y and Discret e Mathematic s

The Victo r Kle e Festschrif t

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DIMACS Series i n Discret e Mathematic s

and Theoretica l Compute r 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 consortiu m o f Rutger s University , Princeto n University ,

AT&T Bel l Labs , Bellcor e

https://doi.org/10.1090/dimacs/004

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

To orde r throug h AM S contac t the AM S Custome r Service s Department , P.O . Bo x 6248 , Providence , Rhod e Islan d 02940-624 8 USA . Fo r VIS A or MASTERCAR D order s cal l 1-800-321 -4AMS . Orde r cod e DIMACS/4 .

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Foreword

This DIMACS volume, The "Victor Klee 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 hi s mathematical contributions . W e especially thank th e editors , Pete r Gritzman n 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 discret e mathematics .

Daniel Gorenstein , Directo r Robert Tarjan , Co-Directo r Fred S . Roberts, Associate Directo r

Brief Contents

Preface i x

Biography o f Victo r Klee x i

Bibliography o f Victor Klee xvi i

Contents (i n alphabetica l orde r by author) xxx i

List o f Papers (b y subjects) xxx v

Contributed Article s 1-60 8

Preface

This volume comprises a collection of research articles dedicated to Victor Klee on the occasion of his 65th birthday i n 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 broa d interests , a wide range o f area s in mathematic s an d it s applications ar e touched upo n here. Thes e areas includ e

• Discrete and Computationa l Geometry , • Classical and Computationa l Convexity , • Convex Polytopes and thei r Relatives , • Combinatorics, Polyhedra l Combinatorics , an d Grap h Theory , • Functional Analysis , • Mathematical Programmin g an d Optimization , an d • Theoretical Compute r Science . Victor Klee has made significant contribution s no t 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 biologica l 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 le t hi s vit a 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 thi s volume and t o give easy access to papers belonging to the same area, we include a list o f papers by subjec t area .

We are indebted t o the Cente r fo r Discret e Mathematic s an d Theoretica l Computer Science , in particular to its director Daniel Gorenstein , an d to the American Mathematica l Societ y fo r thei r hel p i n arrangin g th e publicatio n of thi s volume. W e wish t o than k th e referee s fo r thei r invaluabl e hel p an d the authors for thei r enthusiastic suppor t throughout thi s 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 Victo r Kle e for al l that h e has given to us.

Peter Gritzman n 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, Universit e d e Liege, 198 4 D.Sc, Pomon a 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 an d Technology, 198 2

Alexander von Humbold t 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 an d 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 VICTOR KLE E

Adjunct Professo r o f Compute r 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 it s 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 Postdoctora l 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, Universit y o f Virginia , 1948-194 9 Du Pon t Predoctoral , Universit y o f Virginia , 1945-194 7

Part-time Consultan t W. H. Freeman an d Company , 1976-presen t Holt, Rinehar t an d Winston , Inc. , 1966-197 6 E. I. du Pon t 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 VICTOR 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 th e 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 fo r 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 , Mathematica l Programmin g Society , Operation s Research Society , and Ph i Beta Kappa .

Invited Lecture s International Congres s o f Mathematicians , Vancouver , 197 4 Eighth Internationa l Symposiu m o n Mathematica l Programming , Stan -

ford, 197 3 (Plenary Speaker ) Invited lecture s a t various annua l meeting s o f American Mathemati -

cal Society , Mathematica l 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, Deutsch e Gesellschaf t fii r Mathematik , Okonomi e un d Op-erations Researc h

Invited hou r addresse s a t nationa l o r international conference s de -voted t o the following subjects :

Applications o f Combinatoric s i n the Biologica l and Socia l Science s (1988)

Applied Linea r Algebra (1988 ) Combinatorics an d Geometr y (1989 ) Discrete Optimizatio n (1981 , 1980 , 1977 ) Discrete Geometr y (1981 , 1966 , 1962 ) Convexity (1980 , 1975 , 1965 , 1961 ) Information Linkag e between Applie d Mathematic s an d Industr y (1978)

xiv BIOGRAPH Y O F VICTOR KLE E

Combinatorial Mathematic s (1990 , 1978 , 1969 , 1969 , 1968 , 1968, 1963)

Computing in Algebra and Numbe r Theor y (1975 ) Mathematical Programmin g (1988 , 1982 , 1973 , 1967 ) Set-Theoretic Topology (1973 , 1961 , 1955) Mathematical Method s o f Economic s (1972 ) Graph Theor y an d it s Applications (1983 , 1972 , 1969 ) Algorithmic and Applied Combinatoric s (1986 , 1983 , 1971 , 1969) Teaching of Geometry (1990 , 1988 , 1969 , 1967 ) Calculus of Variations an d Contro l Theory (1968 ) Mathematics o f the Decision Science s (1967 ) Scientific Computin g (1964 ) Functional Analysi s (1964 , 1960 , 1960 ) Polytopes and Conve x Set s (1990 ) Applied an d Computationa l 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 Combinatoria l Theor y Linear Algebra and Application s Mathematics o f Operation s Researc h Discrete and Computationa l Geometr y

Ph.D. Student s 29 in Mathematic s 1 in Applied Mathematic s 1 in Compute r Scienc e

Some othe r activities o f the las t six year s Institute fo r Mathematic s an d it s 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 Discret e an d Computationa l Geometr y Advisory Committee , Progra m i n Applied Linea r Algebra Board o f Governor s

Mathematical Science s Research Institut e Board o f Trustee s

BIOGRAPHY O F VICTOR 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 Mathematica l Societ y

Nominating Committe e Centennial Fellowshi p Committe e Organizing Committee , Summe r Workshop on Mathematica l Devel -opments Relate d t o Linear Programmin g

Invited speake r a t specia l session s on differentia l equations , func -tional analysis , convex sets , combinatorics, discret e geometr y

Mathematical Associatio n o f Americ a Committee o n the Annual Award fo r 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 computationa l geometr y

Operation Researc h Societ y o f Americ a Arranged specia l session o f paper s on mathematica l aspect s o f lin -ear programmin g

Bibliography of 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 VICTO R 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, Duk e 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 Theoreti c Topology , Madison , Wis -consin, Amer . Math . Soc , Providence , pp . 132-134 .

BIBLIOGRAPHY O F VICTO R 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 Functiona l 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 VICTO R 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 OF 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 par t 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 Symposiu m o n Combinatoria l 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 Geometr y 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 VICTO R 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) , Pro c 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 an d M . Juncosa), Math. Ann . 183 , 105-114 .

XXIV BIBLIOGRAPHY OF 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 Mathematica l 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, Combinatoric s (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-minut e color film and 27-page viewer's manual, Individua l Lecture Film Projec t o f the Mathematica l 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 VICTO R 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 , an d 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 Internationa l 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 OF 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, Polyhedra l 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 OF VICTOR KLEE 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 versio n 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 an d Appl . 37 , 277-284 .

1983

Chinese translation of paper 17 6 above appears in Applied Mathematic s and Mathematic s o f Computatio n (People' s Republi c 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 VICTO R 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. Contro l 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 an d 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) , Discret e 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

Contents

(in alphabetica l orde r b y author )

A Dua l Fores t Algorith m fo r th e Assignmen t 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 wit h 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 G I L KALA I 8 1

Computing th e Conve x Hul l i n th e 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 th e 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 Rando m Polytop e i n a Conve x Se t

L. DALL A AN D D . G . LARMA N 17 5

Bodies o f Constan t Widt h i n Riemannia n 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 W O O D 19 3

The Symmetrie s o f th e 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 Hyperplan e 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 , an d th e Grap h o f Nearl y 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 wit h Circle s

G. FEJE S T O T H AN D L . FEJE S T O T H 27 5

Symmetric Solution s t o Isoperimetri c Problem s fo r Polytope s

P. FILLIMA N 28 9

A Globa l Newto n Metho d

A. A . GOLDSTEI N 30 1

Volume Approximatio n o f Conve x Bodie s b y Circumscribe d Polytope s

PETER M . GRUBE R 30 9

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

Convex Minimizer s o f Variationa l Problem s

ERHARD H E I L 32 5

Flattening a Roote d Tre e

PAUL HILFINGER , EUGEN E L . LAWLER , AN D GUNTE R R O T 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 Gracefu l (Bu t Som e ar e Mor e 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 Diamete r o f Graph s o f Conve x Polytope s an d /-Vecto r Theor y G I L KALA I 38 7

Multiply Perspectiv e Simplices , Desmi c Triad s and th e Edelstei n Theorem s

L . M . K E L L Y 41 3

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

Finite Union s o f Close d Subgroup s o f th e ^-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 th e Numbe r o f Antipoda l o r Strictl y Antipoda l 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 Numbe r o f Face s i n ^-Dimensiona l Voronoi Diagram s

RAIMUND SEIDE L 51 7

Stretchability o f Pseudoline s i s NP-Har d PETER W . S H O R 53 1

A Zonotop e 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 Combinatoric s 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 . Ivi c WEIS S 49 3

A Zonotop e 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 ^-star s 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


H. EDELSBRUNNE R AN D M . SHARI R 25 3

Point Set s wit h 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 , Desmi c Triad s and th e Edelstei n Theorem s

L . M . K E L L Y 41 3

On th e Numbe r o f Antipoda l o r Strictl y Antipoda l Pair s o f Point s i n Finite Subset s i n R d


Almost Orthogona l Line s i n E d

M. ROSENFEL D 48 9

Combinatorial Optimizatio n

A Dua l Fores t Algorith m fo r th e Assignmen t Proble m H. ACHATZ , P . KLEINSCHMIDT , AN D K . PAPARRIZO S 1

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


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 C H . SCHUL Z 42 3

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

Computational Geometr y

Computing th e Conve x Hul l i n th e 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


H. EDELSBRUNNE R AN D M . SHARI R 25 3

Exact Uppe r Bound s fo r th e Numbe r 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 Propertie s 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 of 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 Solution s to Isoperimetric Problem s fo r Polytope s P. FILLIMA N 28 9

Volume Approximation o f Convex Bodies by Circumscribed Polytope s P. M . GRUBE R 30 9

Convex Minimizer s o f Variationa l Problem s E. HEI L 32 5

Centers and Invarian t 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 Antipoda l o r Strictl y Antipoda l Pair s o f Point s i n Finite Subset s in Rd


Convexity in General Spaces , Generalized Convexit y

Bodies of Constan t Widt h i n Riemannian Manifold s and Space s of Constan t 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 Congruen t Cylinder s an d with Congruen t Ellipsoid s


On Generi c Globa l 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 o n 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 Polynomial s o n Banach Lattice s

K. SUNDARESA N 57 1

Graph Theory

Recognizing Propertie s o f Periodic Graph s E. COHE N AN D N . MEGIDD O 13 5

Gaps in 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

Every Tree i s Graceful (Bu t Som e Are More Gracefu l tha n Others ) T. C . H u AN D A . B . KAHN G 35 5

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

Mathematical Programmin g

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


A Global Newto n Metho d A. A . GOLDSTEI N 30 1


R. HORS T AN D H . TU Y 34 1

PAPERS B Y SUBJECT xxxi x

Centers an d Invarian t Point s o f Convex 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 Permanen t 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 and Global Optimization

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


R. HORS T AN D H . TU Y 34 1

Packing and Covering

Packing Euclidean Spac e with Congruen t Cylinder s an d with Congruen t Ellipsoid s


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 an d 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 Polyhedra l Map s D. W . BARNETT E 6 3

Some Regular Maps and Thei r Polyhedra l Realization s H. S . M . COXETE R AN D G . C . SHEPHAR D 15 7

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

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

Finite Union s 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 Uppe r Bound s for th e Number o f Faces in ^-Dimensiona l Voronoi Diagram s

R. SEIDE L 51 7

Rigidity Theory and Splines

On Generic Globa l Rigidit y R. CONNELL Y 14 7

The Combinatorics o f Bivariate Spline s W. WHITELE Y 58 7

Stochastic Geometry

Computing the Convex Hul l in the Euclidea n 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 Minimizer s o f Variationa l Problem s E. HEI L 32 5

applied geometry and discrete mathematics · 2019-02-12 · fields, but also to mathematics...

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