Discrete Geometry

DISCRETE GEOMETRY In Honor of W. Kuperberg's 60th Birthday

edited

by

Andras Bezdek Auburn University Auburn, Alabama, U.S.A. Alfred Renyi Institute of Mathematics Hungarian Academy of Sciences Budapest, Hungary

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K Yano, Integral Formulas in Riemannian Geometry (1970) S Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings (1970) V S Vladimirov, Equations of Mathematical Physics (A Jeffrey, ed , A Littlewood, trans) (1970) B N Pshemchnyi, Necessary Conditions for an Extremum (L Neustadt, translation ed , K Makowski, trans ) (1971) L Nanci et al, Functional Analysis and Valuation Theory (1971) S S Passman, Infinite Group Rings (1971) L Domhoff, Group Representation Theory Part A Ordinary Representation Theory Part B Modular Representation Theory (1971, 1972) W Boothbyand G L Weiss, eds, Symmetric Spaces (1972) V Matsushima, Differentiable Manifolds (E T Kobayashi, trans ) (1972) L E Ward, Jr, Topology (1972) A Babakhanian, Cohomological Methods in Group Theory (1972) R Gilmer, Multiplicative Ideal Theory (1972) J Yeh, Stochastic Processes and the Wiener Integral (1973) J Barros-Neto, Introduction to the Theory of Distributions (1973) R Larsen, Functional Analysis (1973) K Yano and S Ishihara, Tangent and Cotangent Bundles (1973) C Procesi, Rings with Polynomial Identities (1973) R Hermann, Geometry, Physics, and Systems (1973) N R Wallach, Harmonic Analysis on Homogeneous Spaces (1973) J Dieudonne, Introduction to the Theory of Formal Groups (1973) / Vaisman, Cohomology and Differential Forms (1973) B -Y Chen, Geometry of Submamfolds (1973) M Marcus, Finite Dimensional Multilinear Algebra (in two parts) (1973, 1975) R Larsen, Banach Algebras (1973) R O Kujala and A L Vitter, eds, Value Distnbution Theory Part A, Part B Deficit and Bezout Estimates by Wilhelm Stoll (1973) K B Stolarsky, Algebraic Numbers and Diophantme Approximation (1974) A R Magid, The Separable Galois Theory of Commutative Rings (1974) B R McDonald, Finite Rings with Identity (1974) J Satake, Linear Algebra (S Koh et al, trans ) (1975) J S Golan, Localization of Noncommutative Rings (1975) G K/ambauer, Mathematical Analysis (1975) M K Agoston, Algebraic Topology (1976) K R Goodeart, Ring Theory (1976) L £ Mansfield, Linear Algebra with Geometric Applications (1976) N J Pullman, Matrix Theory and Its Applications (1976) B R McDonald, Geometric Algebra Over Local Rings (1976) C W Groetsch, Generalized Inverses of Linear Operators (1977) J £ Kuczkowski and J L Gersting, Abstract Algebra (1977) C O Chnstenson and W L Voxman, Aspects of Topology (1977) M Negate, Field Theory (1977) R L Long, Algebraic Number Theory (1977) W F Pfeffer, Integrals and Measures (1977) R L WheedenandA Zygmund, Measure and Integral (1977) J H Curtiss, Introduction to Functions of a Complex Variable (1978) K Hrbacek and T Jech, Introduction to Set Theory (1978) W S Massey, Homology and Cohomology Theory (1978) M Marcus, Introduction to Modern Algebra (1978) E C Young, Vector and Tensor Analysis (1978) S B Nad/er, Jr,Hyperspaces of Sets (1978) S K Segal, Topics in Group Kings (1978) ACM van ROOIJ, Non-Archimedean Functional Analysis (1978) L Command R Szczarba, Calculus in Vector Spaces (1979) C Sadosky, Interpolation of Operators and Singular Integrals (1979) J Cronm, Differential Equations (1980) C W Groetsch, Elements of Applicable Functional Analysis (1980)

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113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171

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172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225

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226. R. Li et a/., Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods (2000) 227. H. Li and F. Van Oysfaeyen, A Primer of Algebraic Geometry (2000) 228. R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applications, Second Edition (2000) 229. A B. Kharazishvili, Strange Functions in Real Analysis (2000) 230. J. M. Appell et a/., Partial Integral Operators and Integra-Differential Equations (2000) 231. A. I. Prilepko et a/., Methods for Solving Inverse Problems in Mathematical Physics (2000) 232. F. Van Oysfaeyen, Algebraic Geometry for Associative Algebras (2000) 233. D. L Jagerman, Difference Equations with Applications to Queues (2000) 234. D. R. Hankerson et a/., Coding Theory and Cryptography: The Essentials, Second Edition, Revised and Expanded (2000) 235. S. Dascalescu et a/., Hopf Algebras: An Introduction (2001) 236. R. Hagen et a/., C*-Algebras and Numerical Analysis (2001) 237. Y. Talpaert, Differential Geometry: With Applications to Mechanics and Physics (2001) 238. R H. Villarreal, Monomial Algebras (2001) 239. A. N. Michel et a/., Qualitative Theory of Dynamical Systems: Second Edition (2001) 240. A. A. Samarskii, The Theory of Difference Schemes (2001) 241. J. Knopfmacher and W -B. Zhang, Number Theory Arising from Finite Fields (2001) 242. S. Leader, The Kurzweil-Henstock Integral and Its Differentials (2001) 243. M. Biliotti et al, Foundations of Translation Planes (2001) 244. A. N. Kochubei, Pseudo-Differential Equations and Stochastics over Non-Archimedean Fields (2001) 245. G. Sierksma, Linear and Integer Programming: Second Edition (2002) 246. A. A. Martynyuk, Qualitative Methods in Nonlinear Dynamics: Novel Approaches to Liapunov's Matrix Functions (2002) 247. B. G. Pachpatte, Inequalities for Finite Difference Equations (2002) 248. A. N. Michel and D. Liu, Qualitative Analysis and Synthesis of Recurrent Neural Networks (2002) 249. J. R. Weeks, The Shape of Space: Second Edition (2002) 250. M. M. Rao and Z. D. Ren, Applications of Orlicz Spaces (2002) 251. V. Lakshmikantham and D. Trig/ante, Theory of Difference Equations: Numerical Methods and Applications, Second Edition (2002) 252. T. Albu, Cogalois Theory (2003) 253. A. Bezdek, Discrete Geometry (2003) Additional Volumes in Preparation

PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS Earl J. Taft Rutgers University New Brunswick, New Jersey

Zuhair Nashed University of Central Florida Orlando, Florida

EDITORIAL BOARD M. S. Baouendi University of California, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology

Anil Nerode Cornell University Donald Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University

S. Kobayashi University of California, Berkeley

David L. Russell Virginia Polytechnic Institute and State University

Marvin Marcus University of California, Santa Barbara

Walter Schempp Universitdt Siegen

W. S. Massey Yale University

Mark Teply University of Wisconsin, Milwaukee

PREFACE

It is with great pleasure that we publish this collection of papers dedicated to Professor Wlodzimierz Kuperberg commemorating the occasion of his sixtieth birthday, January 19, 2001. All included papers were received in the second half of 2001 and contain new results. The principal theme of this volume is intuitive geometry. This book covers packing and covering theory, tilings, combinatorial geometry, convexity, and computational geometry. These topics belong to subjects on which Professor Kuperberg has made strong impacts and is continuing to make many deep contributions. A special feature of the volume is a problem collection, which is an extended version of the set of problems accumulated at the Discrete Geometry Special Session, organized by Andras Bezdek, and held at the Annual Joint Mathematics Meetings in New Orleans. I would like to express my gratitude to the authors for contributing to this anniversary volume and to the many referees who gave their valuable time to read the manuscripts. I also thank Professor Krystyna Kuperberg, who provided several of the biographical details for the introductory paper. Andras Bezdek

CONTENTS

Preface Contributors Biographical notes and work ofW. Kuperberg Andras Bezdek and Gabor Fejes Toth

1. Transversal lines to lines and intervals Jorge L. Arocha, Javier Bracho, and Luis Montejano

in ix xiii

1

2. On a shortest path problem of G. Fejes Toth Donald R. Baggett and Andras Bezdek

19

3. A short survey of (r, ^-structures Vojtech Bdlint

27

4. Lattice points on the boundary of the integer hull Imre Bdrdny and Kdroly Boroczky, Jr.

33

5. The Erdos-Szekeres problem for planar points in arbitrary position Tibor Bisztriczky and Gabor Fejes Toth

49

6. Separation in totally-sewn 4-polytopes Tibor Bisztriczky and Deborah Oliveros

59

7. On a class of equifacetted polytopes Gerd Blind and Roswitha Blind

69

8. Chessboard Ramsey numbers Jens-P. Bode, Heiko Harborth, and Stefan Krause

79

9. Maximal primitive fixing systems for convex figures Vladimir Boltyanski and Herndn Gonzdlez-Aguilar

85

CONTENTS

10. The Newton-Gregory problem revisited Kdrohj Boroczky

103

11. Arrangements of 13 points on a sphere Kdroly Boroczky and Ldszlo Szabo

111

12. On point sets without k collinear points Peter Brass

185

13. The Beckman-Quarles theorem for rational d-spaces, d even and d > 6 Robert Connelly and Joseph Zaks

193

14. Edge-antipodal convex polytopes - a proof of Talata's conjecture 201 Baldzs Csikos 15. Single-split tilings of the sphere with right triangles Robert J. MacG. Dawson

207

16. Vertex-unfoldings of simplicial manifolds Erik D. Demaine, David Eppstein, JeffErickson, George W. Hart, and Joseph O'Rourke

215

17. View-obstruction through trajectories of co-dimension three Vishwa C. Dnmir and Rajinder J. Hans-Gill

229

18. Fat 4-polytopes and fatter 3-spheres David Eppstein, Greg Kuperberg, and Giinter M. Ziegler

239

19. Arbitrarily large neighborly families of congruent symmetric convex 3-polytopes JeffErickson and Scott Kim

267

20. On the non-solidity of some packings and coverings with circles August Florian and Aladdr Heppes

279

21. On the mth Petty numbers of normed spaces Kdrohj Bezdek, Marion Naszodi and Baldzs Visy

291

22. Cubic polyhedra Chaim Goodman-Strauss and John M. Sullivan

305

CONTENTS

23. "New" uniform polyhedra Branko Griinbaum 24. On the existence of a convex polygon with a specified number of interior points Kiyoshi Hosono, Gyula Kdrolyi, and Masatsugu Urabe 25. On-line 2-adic covering of the unit square by boxes Janusz Januszewski and Marek Lassak 26. An example of a stable, even order quadrangle which is determined by its angle function Jdnos Kineses

331

351

359

367

27. Sets with a unique extension to a set of constant width Marion Naszodi and Baldzs Visy

373

28. The number of simplices embracing the origin Jdnos Pack and Mario Szegedy

381

29. Helly-type theorems on definite supporting lines for /c-disjoint families of convex bodies in the plane Sorin Revenko and Valeriu Soltan

387

30. Combinatorial aperiodicity of polyhedral prototiles Egon Schulte

397

31. Sequences of smoothed polygons G. C. Shephard

407

32. On a packing inequality by Graham, Witsenhausen and Zassenhaus Jorg M. Wills

431

33. Covering a triangle with homothetic copies Zoltdn Fu'redi

435

34. Open Problems Andrds Bezdek

447

Index

459

CONTRIBUTORS

Jorge L. Arocha Mathematical Institute, Universidad Nacional Aut6noma de Mexico , Mexico City, Mexico Donald R. Baggett Auburn University, Auburn, Alabama Vojtech Balint University of Zilina, Zilina, Slovak Republic Imre Barany Renyi Institute of Mathematics of the Hungarian Academy of Sci., Budapest, Hungary Andras Bezdek Auburn University, Auburn, Alabama and Renyi Institute of Mathematics of the Hungarian Academy of Sciences, Budapest, Hungary Karoly Bezdek Eotvos Lorand University, Budapest, Hungary Tibor Bisztriczky University of Calgary, Calgary, Canada Gerd Blind University of Stuttgart, Stuttgart, Germany Rozwitha Blind Waldburgstr. 88, D 70563 Stuttgart, Germany Jens-P. Bode Technical University, Braunschweig, Germany Vladimir Boltyanski Mathematical Research Center, Guanajuato, Mexico Karoly Boroczky Eotvos Lorand University, Budapest, Hungary Karoly Boroczky Jr. Alfred Renyi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary Javier Bracho Mathematical Institute, Universidad Nacional Aut6noma de Mexico , Mexico City, Mexico Peter Brass Free University Berlin, Berlin, Germany Robert Connelly Cornell University, Ithaca, New York

x

CONTRIBUTORS

Balazs Csikos Eotvos Lorand University, Budapest, Hungary Robert J. MacG. Dawson Saint Mary's University, Halifax, Canada Erik D. Demaine MIT Laboratory for Computing Science, Cambridge, Massachusetts Vishwa C. Dumir Centre for Advanced Study in Mathematics, Panjab University, Chandigarh, India David Eppstein University of California, Irvine, California Jeff Erickson University of Illinois at Urbana-Champaign, Urbana, Illinois Gabor Fejes Toth Alfred Renyi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary August Florian University of Salzburg, Salzburg, Austria Zoltan Fiiredi University of Illinois at Urbana-Champaign, Urbana, Illinois and Alfred Renyi Institute of Math., Hungarian Academy of Sci., Budapest, Hungary Hernan Gonzalez-Aguilar Mathematical Research Center, Guanajuato, Mexico Chaim Goodman-Strauss University of Arkansas, Fayetteville, Arkansas Branko Griinbaum University of Washington, Seattle, Washington Rajinder J. Hans-Gill Centre for Advanced Study in Mathematics, Panjab University, Chandigarh, India Heiko Harborth Technical University of Braunschweig, Braunschweig, Germany George W. Hart

SUNY Stony Brook, Stony Brook, New York

Aladar Heppes Alfred Renyi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary Kiyoshi Hosono Tokai University, Shimizu, Shizuoka, Japan Janusz Januszewski Academy of Technical and Agricultural Sciences, Bydgoszcz, Poland Gyula Karolyi Eotvos Lorand University, Budapest, Hungary Scott Kim P.O.Box 2499, El Granada, California 94018

CONTRIBUTORS

xi

Janos Kineses Szeged University, Szeged, Hungary Stefan Krause Technical University of Braunschweig, Braunschweig, Germany Greg Kuperberg Department of Mathematics, University of California, Davis, California Lassak Marek Academy of Technical and Agricultural Sciences, Bydgoszcz, Poland Luis Montejano Mathematical Institute, UNAM, Mexico City, Mexico Marton Naszodi Eotvos Lorand University, Budapest, Hungary Joseph O'Rourke Smith College, Northampton, Massachusetts Deborah Oliveros University of Calgary, Calgary, Canada Janos Pach Courant Institute, New York, New York and Renyi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary Sorin Revenko George Mason University, Fairfax, Virginia Egon Schuite Northeastern University, Boston, Massachusetts Geoffrey C. Shephard University of East Anglia, Norwich, England Valeriu Soltan George Mason University, Fairfax, Virginia John M. Sullivan University of Illinois at Urbana-Champaign, Urbana, Illinois Laszlo Szabo Computer and Automation Research Institute of the Hungarian Academy of Sciences, Budapest, Hungary Mario Szegedy Rutgers University, New Brunswick, New Jersey Masatsugu Urabe Tokai University, Shimizu, Shizuoka, Japan Balazs Visy Eotvos Lorand University, Budapest, Hungary Jorg M. Wills University of Siegen, Siegen, Germany Joseph Zaks University of Haifa, Haifa, Israel Giinter M. Ziegler Technical University of Berlin. Berlin. Germany

BIOGRAPHICAL NOTES AND WORK OF W. KUPERBERG

Wiodzimierz (Wlodek) Kuperberg was born on January 19th, 1941, in Belarus, just outside the Polish border. His mother, father and two older siblings abandoned their home in Warsaw and headed east, escaping the horrors of WWII. In 1946, the family consisting of the parents and four children returned to Poland and made its home in the city of Szczecin. In 1959, Kuperberg enrolled at Warsaw University as a student of mathematics. His original intention was to study physics, but he changed his mind after winning the Mathematics Olympiad, the most prestigious math competition in Poland for high school students. Surprised by this sudden success and fascinated by the mathematical charm of Kazimierz Kuratowski, who spoke at the olympiad ceremony, Wiodzimierz was hooked on math. The mathematics program at Warsaw University had much to offer. The students were exposed to open problems early in seminars, lectures, and informal conversations. As a freshman, Wlodek Kuperberg, together with his good friend Wlodek Holsztyriski, a sophomore, solved a problem in geometry and published the paper titled "On a property of tetrahedra" in Wiadomosci Matematyczne (Annals of the Polish Mathematical Society) 6 (1962), 13-16. Geometry was a natural choice of study. Karol Borsuk, who had a strong influence at Warsaw University, was known for several famous conjectures in this area. In those years, however, Borsuk devoted his seminars to topology: theory of retracts and shape theory. As for many of Borsuk's students, topology became Kuperberg's research area. As a graduate student, he wrote three important papers in topology: "Stable points of a polyhedron" Fund. Math. 59 (1966), 43-48; "A cyclic two-dimensional compactum which contains no irreducibly cyclic two-dimensional subcompactum" Doklady Akad. Nauk USSR 182 (1968), 35-37; and "Homotopically labile points of locally compact metric spaces" Fund. Math. 73 (1971/72), 133-136. The second paper has a nice history. Kuperberg solved a problem posed by P.S. Alexandroff and sent a hand-written paper (in Russian) to him asking whether the problem was still open. On Alexandroff's invitation, the paper appeared in the Doklady.

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BIOGRAPHICAL NOTES

Kuperberg received his PhD degree at Warsaw University in 1969. He wrote his dissertation in algebraic topology under Borsuk's direction, inspired by Borsuk's geometric style in topology and by the works of H. Hopf and L. Vietoris. In addition to Borsuk, other great mathematicians such as Stan Mazur, Kazimierz Kuratowski and Waclaw Sierpiriski, who at that time were active members of the Warsaw mathematical community, had a strong influence on Kuperberg's research. During his short period of employment at Warsaw University, Kuperberg received an Excellence in Teaching and Research Award given by the university, and the Polish Mathematical Society Award for Young Mathematicians. He also wrote 3 high school textbooks, one of which was a geometry book that became widely used in Poland. The 1960's and 1970's were very eventful years. In 1964, Wlodek Kuperberg married Krystyna Trybulec, a student of mathematics, and their son Greg was born in 1967. Greg is now a math professor at the University of California at Davis. In 1969, the Kuperbergs hastily moved to Sweden. Soon after their arrival their second child, Anna, was born, who is now a photo journalist in San Francisco. Wlodek worked at Stockholm University for two and a half years, lecturing in Swedish, a language he had to learn quickly. During that time he submitted the results of his dissertation for publication, his 8th paper, "On certain homological properties of finite-dimensional compacta. Carriers, minimal carriers and bubbles" Fund. Math. 83 (1973), 7-23. Together with Krystyna, he wrote a problem set in topology. With the proximity of the Mittag-Leffler Institute and two other universities, Uppsala University and the Royal Institute of Technology in Stockholm, the mathematical environment was rich. To Wlodek, however, a big step forward was attending a topology conference in Houston, Texas, in 1971, and assuming a two-year visiting position at the University of Houston in 1972. There he worked in continuum theory, collaborating with Andrew Lelek and Howard Cook. In "Mapping arcwise connected continua onto cyclic continua", Colloq. Math. 31 (1974), 199-202, he defines an algebraic invariant useful in the study of continua. In 1974, Wlodek and his wife Krystyna, who received a PhD degree at Rice University in Houston that year, took employment at the math department at Auburn University, a land-grant institution in Auburn, Alabama. Krystyna is known for her results in topology and dynamical systems, in particular for constructing a smooth counterexample to the Seifert conjecture. Wlodek continued his research in topology, returning to the geometric topics related to his work in Poland. His paper "Unstable sets and the set of unstable points", Bull. Acad. Polon. Sci. Sr. Sci. Math. Astronom. Phys. 26 (1978), 281-285, is a good example. He was also interested in shape theory, which was developed and popularized by Borsuk. Wlodek received tenure in 1977 and was made Professor in 1982.

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Time spent at math conferences has always been productive and enjoyable. At the 1973 Topology Conference in Blacksburg, Virginia, Wlodek "earned" a dinner from R.H. Bing by answering a question that Bing posed during the conference. Bing included the solution in his paper titled "An unusual map of a 3-cell onto itself. The picture enclosed here shows Bing's note written on the back of the banquet dinner ticket. Another remarkable conference took place in Denton, Texas, in 1979. It was dedicated to the Scottish Book, a set of problems collected in Lwow before WWII. One of the main contributors to the book, Stan Ulam, was in Denton and Kuperberg answered the Scottish Book Problem 110 which was posed by Ulam in 1935. At the Denton conference, Kuperberg gave the first counterexample, a 1-dimensional continuum with certain properties. Subsequent counterexamples based on flows in manifolds by P. Mine, K. Kuperberg, and C.S. Reed led to important research in dynamical systems. In the early 1980's, influenced by the work of Vladimir Boltianski, Don Chakerian, and Laszlo Fejes Toth, W. Kuperberg revived his interest in geometry with the paper "Packing convex bodies in the plane with density greater than 3/4", Geom. Dedicata 13 (1982), 149-155. This result led to an invitation to a geometry conference in Siofok, Hungary, and subsequently to a very fruitful collaboration with the geometers in Hungary. Both authors of this article profited a lot of the collaboration with Wlodek. On the meeting in Siofok Wlodek showed once more, but not for the last time, that his best ideas come in the inspiring environment of a meeting: The sharp inequality claiming that the quotient of the packing density and the covering density of a plane convex body is at least 3/4 (Geometriae Dedicata 13 (1982), 149-145) was proved there. Kuperberg together with Andras Bezdek - who joined the department of mathematics at Auburn University in 1991 - studied "Maximum density space packing with congruent circular cylinders of infinite length" Mathematika (37) 1990, 74-80. Prior to this paper no convex solid S in E3 (bounded or not) with 6(S) ^ I had its packing density computed explicitly. Jointly with A. Bezdek and E. Makai, Kuperberg generalized Gauss' theorem by proving that, if a sphere packing in E3 consists of parallel strings of spheres, then the density of the packing is smaller than or equal to ?r/\/l8. In "Applied geometry and discrete mathematics", DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 4 (1991), 71-80, Kuperberg and A. Bezdek produced a number of results concerning packings of compact convex cylinders in .E3 and ellipsoids. Concerning ellipsoids, a surprise was discovered: certain congruent ellipsoids in Ed (for d > 3) can be packed in a nonlatticelike manner more densely than in a lattice-like manner. In view of Hales'

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proof of Kepler's conjecture this means that in E3 packings of congruent replicas of certain ellipsoids can achieve higher density than any packing of equal balls. In the subsequent years Gabor Fejes Toth also became Wlodek's good friend and a frequent visitor in Auburn. Inspired by the result on dense ellipsoid packings, they solved the dual problem for coverings in "Thin nonlattice covering with an affine image of a strictly convex body", Mathematika 42 (1995), 239-250. Gabor is also very proud of their surveys, in particular of the paper "Packing and covering with convex sets" in Handbook of Convex Geometry, 1993, 799-860. Wlodek quite often uses his background of topology in his research in geometry. The paper on "On-line covering a cube by a sequence of cubes", Discrete Comput. Geom. 12 (1994), 83-90 is one example. The applied on-line covering method uses a cube-filling curve which is an analogue of the classical Peano curve. Another result of topological flavor is his elaborate construction in the paper "Knotted lattice-like space fillers", Discrete Comput. Geom. 13 (1995), 561-567. Given any handlebody (ball with handles attached), such that the handles may be knotted and/or linked with one another, and to which mutually disjoint tunnels have been drilled out (also potentially knotted and linked), this also forms the topological type of a tile that can be used to tile Euclidean 3-space. Quite often Wlodek's proofs "come straight from the book". As a recent example we mention his paper "Holey coronas - a solution of the GriinbaumShepard conjecture", American Mathematical Monthly 17 (2000), 551-555. In this paper he disproves the conjecture that the corona of every tile in an isohedral tiling of the plane by convex tiles is simply connected. He gives a very elegant construction to obtain a tiling in which every corona is a handlebody with an arbitrarily large number of handles. We already praised Wlodek as an excellent problem solver. We have to mention that he is an equally good problem raiser. A further special strength of his is clarity of presentation. Difficult tricks appear natural as he explains them. Take for example his joint paper with his son Greg, which presents the best known lower bound for the packing density of a convex plane body. The second named author of this article remembers refereeing this paper. He read the paper in one afternoon and sent his supporting report next day. Later the editors asked him whether he was sure that the paper is worth publishing, as the proof appeared to be trivial. Confronted with this question he sat down and tried to reconstruct the proof. Only when this attempt failed did he realize that the seeming triviality was caused by the absolutely clear presentation. The result, of course, was published in Discrete Comput. Geom. 5 (1990), 389-397 and it is widely cited. A whole section with detailed proof is devoted to it in the book entitled "Combinatorial Geometry" by Janos Pach and Pankaj Agarwal. Wlodek's talks show the

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very same clarity as his papers. Because of the style of his presentations he is frequently chosen by conference organizers to deliver the closing speech. In 1996, Wlodek was awarded a five-year Alumni Professorship at Auburn University. He conducted most of his research at Auburn, occasionally spending sabbatical time at another university and frequently attending mathematical meetings. In the Summer of 1999, he received an Erdos Professorship and visited the Renyi Institute of the Hungarian Academy of Sciences in Budapest. There were many geometers around to talk about mathematics. He found himself in a congenial and familiar environment. For example, Gyula Katona, the director of the institute, was an old friend of Wlodek. The friendship originated when they were both high school students and competitors at the first International Mathematics Olympiad. The Kuperbergs have a reputation for exceptional hospitality. Those who visited their house experience a uniquely relaxed atmosphere. This is how the 35 participants of the Hungarian-US Geometry Workshop will remember the welcome party, held in the Kuperbergs' backyard, surrounded by blooming azaleas in the middle of a warm March in 2000. Immediately after proposing the volume to fellow geometers, the editor of this volume received overwhelming support, as shown by comments, some of which is reprinted here: ... I like Wlodek (and his mathematics) a lot, and would be happy to submit an article ... Wlodek is one of the youngest 60-year-olds I know ... it's a wonderful idea to publish this special volume ... it seems hard to believe that he is turning 60 this year! What is that lucky date? ...

Andrds Bezdek and Gdbor Fejes Toth

TRANSVERSAL LINES TO LINES AND INTERVALS

Jorge L. Arocha Institute de Matematicas, UNAM, Ciudad Universitaria, Circuito Exterior, Mexico D.F. 04510, Mexico. Javier Bracho Institute de Matematicas, UNAM, Ciudad Universitaria, Circuito Exterior, Mexico D.F. 04510, Mexico. Luis Montejano Institute de Matematicas, UNAM, Ciudad Universitaria, Circuito Exterior, Mexico D.F. 04510, Mexico.

ABSTRACT. We prove three theorems. A set of lines in RPn has a transversal line if and only if any six of them have a transversal line. The same holds when any five of them have a transversal line, provided that the set of lines is in general position and there are at least seven of them. A finite set of intervals in Rn has a transversal line if and only if any six of them have a transversal line compatible with a given linear order.

1. INTRODUCTION Helly's Theorem reads: let C be a family of compact convex sets in Rn; if every n + I of the sets in C have a common point, then all the family has a common point. Hadwiger showed that an extra hypothesis is needed to prove an analogous theorem for "lines that cross" convex sets in the plane. Hadwiger's Theorem, [7], can be stated as follows. Let {d,C2, ...,Cn} be a finite collection of convex sets in the plane such that for any three, d, Cj,Ck, i < j < k, there is a line crossing them precisely in that order; then there exists a line crossing all the sets in the collection. A fruitful direction in which Hadwiger's Theorem has been generalized is for hyperplane transversals. It was opened by Goodman and Pollack in [5], where they gave necessary and sufficient conditions for the existence of a hyperplane transversal to convex sets in any dimension. That work has i

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been pushed further by them and other authors (see [1],[2] and the references there). However, until now, and as far as we know, no Hadwiger-type theorem is known for transversal line in W1 for n > 2. Some work has been done to obtain criteria for the existence of transversal lines to special classes of convex sets. For example, there is an open conjecture due to Katchalski [9]: if every m members of a collection of pairwise disjoint unit balls in E3 have a transversal line, then the entire collection has a transversal line. For more information we refer the reader to the excellent surveys [3],[4],[6] and [10]. The goal of this paper is to study transversal lines to families of lines and intervals in nth dimensional space (n > 2); we prove a Helly-type theorem for lines and a Hadwiger-type theorem for intervals. Consider lines first. Note that when talking about two lines in W1 that intersect, there is always a limiting case of "intersection at infinity" when they become parallel (and remain coplanar). To avoid awkward argumentations, or changing the concept of transversality for coplanarity, it is better to complete W1 to the real projective n-dimensional space Pn, and simply define that two lines there are transversal if they intersect, and that a line is transversal to a set of lines if it is transversal to each of them. Then all our results will have obvious translations to the affine case. Our opening Theorem is the following. Theorem 1.1. Let C be a collection of lines in Pn. // every 6 of them have a transversal line, then C has a transversal line. It seems strange that no Hadwiger-type assumption is needed, so that by adding the extra condition of partial transversal lines consistent with a given linear or cyclic ordering, one might expect that the "magic number", 6, can be lowered to 5 in the theorem. This is not the case. There are examples of 6 lines with 5 to 5 transversal lines that meet them in a given linear or cyclic ordering, but with no complete transversal. The study of such examples leads to a refinement in a different direction. Namely, the "magic number" can be lowered to 5 if £ contains enough lines (at least 7) and they are in general position (Theorem 5.1 proved in Section 5). The proof of Theorem 1.1 is remarkably easy for the general case (lines in general position). It relies on simple properties of hyperboloids, that is, quadratic surfaces with two line rulings. Most of these facts are well known or straightforward, however, we feel the need to write them down in Section 2 for the benefit of the unaware reader and to establish notation. In Section 3 we prove the degenerate case of Theorem 1.1. In Section 4 we describe the examples of 6 lines with 5 to 5 transversals. Finally, in Section 6 we extend the results to the general case of intervals, rays or lines in W1 (n > 3), proving a Hadwiger-type theorem (Theorem 6.1).

TRANSVERSAL LINES TO LINES

3

2. HYPERBOLOIDS AND THEIR RULINGS Given a family £ of lines in Pn, we will denote by h (£) the set of all the lines transversal to £, and by |/i(£)| C Pn, called its support, the union of all those lines. We will be working with these sets constantly, so that, even if its obvious, it is important to bear in mind that

Our main tool will be the understanding of h(£) for small sets of lines, £ = {^1,^2, ••• ,^fc} with k < 6. For the remainder of this section, we will consider such a set of lines £, for growing fc, with the extra assumption that they are in general position, that is, that no two of them intersect. Until the next section we will address the degenerate case when some lines in £ may meet. Let us begin with k = 2. For any point p% 6 t^, we have a plane passing through t\ and p%, which we denote t\ Vp2- Observe that any point p € t\ Vp2 different from p2 is in a unique line through p% and transversal to l\ (namely, p V pz). As ^2 £ ii varies, the planes i\ V p% span a 3-dimensional flat. Therefore \h(£\,£z}\ = P3, and every point there not in i\ or 1% is in a unique transversal to them (in a unique line in ^(^1,^2))- One can also think of h (^i, ^2) naturally parametrized by the torus i\ x 1% (the projective line is a circle); namely, for each p\ ^ t\, pi G ti, we have the transversal Now, consider a new line. If i\ and £2 have a common transversal with 4 (M4,^2,4) ^ 0),then ^3 intersects the 3-flat |/i(^2)|. But if> moreover, ^1,^2 and £3 have more than one transversal line (tj/i(^i,^2,^s) > 1), then £3 is contained in the 3-flat |/i(^i,^2)| (because it has two points there) and h (^1,^2, ^3) grows to be a projective line: it can be naturally parametrized by the points in £3 where the transversals intersect, because of the uniqueness remark in the preceding paragraph. In this case, three lines in general position in P3, ^(^1,^2,^3) is one ruling (naturally parametrized by -intersection with- either of the three lines) of the hyperboloid \h (^1,^2,^3)!; the unique one that contains the three lines as subsets. For a beautiful exposition of this idea see the opening paragraphs of [8]. The main fact we need about hyperboloids is that |/i(^i,^2,^3)| has another ruling which we call the orthogonal ruling, and denote it h (^1,^2, ^3) , in which the ^ lie. Namely, consider any three (different) lines •£/- , ^ , £j- 6 h (i\,ti ,^3), since they have at least the transversal lines ^1,^2 and ^3, then |/i (•^jS^ is also a hyperboloid, and it happens that 1,^(^1,^2,^3)! = \h (^i",^ Then we define h(^^^}L •= h (££,£%,££) D {^1,^2,^3}- Summarizing, h (£1,^2,^3) is either empty, a unique line or the ruling (parametrized by the

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projective line) of a hyperboloid, which has another ruling containing the 3 lines. The main property we will repeatedly use about hyperboloids is that if a line meets one in more than 2 points, then it belongs to one of its two rulings. This follows, of course, because they are given by quadratic equations, but also from the simple facts we have gadered. For k = 4, suppose that ^(^1,^2,^3,^4) > 2, we want to prove that in this case, -£1,^2,^3 and £4 belong to a ruling of a hyperboloid. By hypothesis we can take three lines t^,^,^ 6 h(t\,t-2.,l$,t£). Sinceft-(^1/2, 4, 4) C h (t\,t 6 by taking new points pi G P, 7 < i < fc, and defining ^ —p\/p^ they maintain 5 to 5 transversals but not all 6 to 6. Example 2. We now avoid the concurrence of many lines in the example ahovp. by taking i^J^J.^. in general position. Let C = \h (£4. £5,