Geometry - Intuitive, Discrete, and Convex: A Tribute to László Fejes Tóth

384 Pages·2013·8.768 MB·English

by  Imre Bárány, Károly Jr. Böröczky, Gábor Fejes Tóth, Janos Pach (eds.)

BOLYAI SOCIETY 24
MATHEMATICAL STUDIES
BOLYYYAI SOCIETY MATHEMATICAL STUDIES
Editor-in-Chieef: Series Editor:
Gábor Fejes Tóth Dezső˝˝ Miklós
Publication Boarrrd:
Gyula O. H. Katona · László Lovász · Péter Pál Pálfffyy
András Recski · András Stipsicz · Domokos Szász
1. Combinatorics, Paul Erdős is Eightyyy, Vol.1
D. Miklós, V.T. Sós, T. Szőnyi (Eds.)
2. Combinatorics, Paul Erdős is Eighty, Vol.2
D. Miklós, V.T. Sós, T. Szőnyi (Eds.)
3. Extremal Problems fffor Finite Sets
P. Frankl, Z. Füredi, G. O. H. Katona, D. Miklós (Eds.)
4. Topology with Applications
A. Császár (Ed.)
5. Approximation Theory and Function Series
PPP. Vértesi, L. Leindler, Sz. Révész, J. Szabados, V. Totik (Eds.)
6. IntuitiveGeometry
I. Bárányyy, K. Böröczkkkyy (Eds.)
7. Graph Theory and Combinatorial Biology
L. Lovász, A. Gyárfás, G. Katona, A. Recski (Eds.)
8. Low Dimensional Topology
K. Böröczkyyy, Jr., W. Neumann, A. Stipsicz (Eds.)
9. Random Walks
P. Révész, B. Tóth(Eds.)
10. Contemporaryy Combinatorics
B. Bollobás (Ed.)
11. Paul Erdős and His Mathematics I+II
G. Halász, L. Lovász, M. Simonovits, V. T. Sós(Eds.)
12. Higher Dimensional Varieties and Rational Points
K. Böröczky, Jr., J. Kollár, T. Szamuely (Eds.)
13. Surgery on Contact 3-Manifffolds and Stein Surfffaces
B. Ozbagci, A. I. Stipsicz
14. A Panorama of Hungarian Mathematics in the Twentieth Century, Vol. 1
J. Horváth (Ed.)
15. More Sets, Graphs and Numbers
E.Győri, G.O. H. Katona,, L. Lovász (Eds.)
16. Entropy, Search, Complexity
I. Csiszár, G. O. H. Katona,, G. Tardos (Eds.)
17. Horizons off Combinatorics
E. Győri, G.O. H. Katona,, L. Lovász (Eds.)
18. Handbook of Large-Scale Random Networks
B. Bollobás, R. Kozma, D. Miklós(Eds.)
19. Building Bridges
M. Grötschel, G. O. H. Katona(Eds.)
20. Fete of Combinatorics and Computer Science
G.O.H. Katona, A. Schriijver, T. Szőnyi (Eds.)
21. An Irregular Mind, Szemerédi is 70
I.Bárányyy, J.Solymosi (Eds.)
22. Cylindric-like Algebras and Algebraic Logic
H. Andréka, M. Ferenczi, I. Németi (Eds.)
23. Deformations of Surface Singularities
A.Némethi, Á. Szilárd (Eds.)
Imre Bárány
Károly J.Böröczky
Gábor FejesTTTóth
János Pach
(Eds.)
Geometry –
Intuitive, Discrete, and Convex
ATTTribute to LászlóFeejesTóth
(cid:2)(cid:3)(cid:4)(cid:5)
JJJÁNOS BOLYYYAI MATHEMATICCCAL SOCIETTTY
Imre Bárány
KárolyyJ. Böröczky
Gábor Fejes Tóth
János Pach
Alfffréd Rényi Institute off Mathematics
Hungarian Academy of Sciences
Reáltanoda u. 13-15
Budapest 1053
Hungary
E-mail: [email protected]
E-mail: [email protected]
E-mail: [email protected]
E-mail: [email protected]
Mathematics Subject Classifffication (2010): 05-06, 52-06
Libraryyof Congress Control Number: 2013950614
ISSN 1217-4696
ISBN 978-3-642-41497-8 Springer Berlin Heidelberg New York
ISBN 978-963-9453-17-3 János Bolyai Mathematical Society, Budapest
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Contents
Contents .......................................................... 5
Preface ............................................................ 7
List of Publications of La´szlo´´ Fejes To´th .................... 9
Akiyama, J., Kobayashi, M., Nakagawa, H., Nakamura, G.
and Sato, I.: Atoms for Parallelohedra ............................ 23
Bezdek, K.: Tarski’s Plank Problem Revisited ...................... 45
Bezdek, A. and Kuperberg, W.: Dense Packing of Space with
Various Convex Solids ............................................. 65
Brass, P.: Geometric Problems on Coverage in Sensor Networks .... 91
Gruber, P. M.: Applications of an Idea of Vorono˘˘ı, a Report ....... 109
Gru¨¨nbaum, B.: Uniform Polyhedrals ................................159
Holmsen, A. F.: Geometric Transversal Theory: T(3)-families in
the Plane ..........................................................187
Montejano, L.: Transversals, Topology and Colorful Geometric
Results ............................................................ 205
Pach, J., Pa´lvo¨¨lgyi, D. and To´´th, G.: Survey on Decomposi-
tion of Multiple Coverings ......................................... 219
Schaefer, M.: Hanani–Tutte and Related Results .................. 259
Schneider, R.: Extremal Properties of Random Mosaics ............ 301
Smorodinsky, S.: Conflict-Free Coloring and its Applications .......331
Preface
In the summer of 2008, the editors of the present volume organized a well
attended conference at the R´enyi Institute of the Hungarian Academy of
SciencestocommemoratethehighlyinfluentialworkandcharacterofL´´aszl´o
Fejes T´oth (1915–2005), one of the best known Hungarian geometers of all
time,whoservedasadirectoroftheInstituteforthirteenyears. Theideaof
publishingacollectionofessaysdedicatedtohimwasconceivedshortlyafter
theconference,anditwasembracedbyanumberofoutstandingcolleagues.
Hungary produced many famous mathematicians in the twentieth cen-
tury,butonlyfewofthemsparkedtheinterestofalargenumberoftalented
students in his subject and created an active school. L´´aszl´o Fejes T´´oth was
one of the exceptions.
Like Paul Erd˝˝os, John von Neumann, and Paul Tur´an, he defended
his thesis under the supervision of Leopold Fej´´er at P´azma´´ny University,
Budapest. In his thesis, he solved a problem in Fourier analysis. He found
the problem and its solution entirely by himself, when reading a classical
monograph of Francesco Tricomi.
After his army service, he started teaching in Kolozsv´ar (Cluj). He
met Dezs˝˝o L´´aza´r, a mathematics teacher at the local Jewish High School,
who told him about an exciting open problem that had a huge impact
on Fejes T´´oth’s later work: How should one arrange n points in the unit
square so as to maximize the minimum distance between them? In other
words, what is the maximum density of a packing of n congruent disks
in a square? Unaware of earlier work by Axel Thue, L´aszl´o Fejes T´´oth
found an asymptotically tight answer to this question. He generalized the
problem in many different directions. A few years later he showed that
the maximum density of a packing of congruent copies of any centrally
symmetricconvexbodyintheplaneisattainedforalatticepacking. Healso
solved another important open problem. Steiner conjectured that among
allconvexpolytopesofunitsurfaceareathatarecombinatoriallyequivalent
to a given Platonic body, the regular polytope has the largest possible
volume. For tetrahedra and octahedra, the conjecture can be easily verified
by symmetrization. Fejes T´oth developed a new technique using sums of
8
Preface
spherical moments to prove Steiner’s conjecture for the cube and for the
dodecahedron. (Theproblemisstillopenfortheicosahedron.) Themethod
has turned out to have many other interesting consequences.
Several years of systematic research in this area resulted in Fejes T´´oth’s
monograph, “Lagerungen in der Ebene, auf der Kugel und im Raum,”
which appeared in the prestigious series of Springer-Verlag, Grundlehren
der mathematischen Wissenschaften in 1953. The book became an instant
classic. As Ambrose Rogers wrote one decade later, “Until recently, the
theory of packing and covering was not sufficiently well developed to justify
the publication of a book devoted exclusively to it. After the publication of
L. Fejes T´oth’s excellent book in 1953, there would be no need for a second
work on the subject, but for the fact that he confines his attention mainly
totwoandthreedimensions.” Thehigherdimensionalproblemsandresults
inspired by Fejes T´oth’s work have led to important discoveries in coding
theory, in combinatorics, and in many other areas. Tom Hales’ solution of
“Kepler’sconjecture,” thespherepackingprobleminthreedimensions, was
motivated by Fejes T´oth’s program outlined in his book. Sixty years after
the publication of its first edition, the Lagerungen is still considered a basic
work in geometry. Its annotated English translation will appear soon.
OneofL´´aszl´oFejesT´´oth’smostimpressiveabilitieswastoaskbeautiful
and deep mathematical questions. Many of these questions can be found
in the short communications listed at the end of Fejes T´oth’s complete
bibliography,includedinthisvolume. Hisworkandhismodest,unassuming
personalityhadalastingimpactontheprofessionallifeanddevelopmentof
the editors as well as many of the contributors of this volume.
Imre B´ar´any
K´´aroly B¨or¨¨oczky, Jr.
G´´abor Fejes T´´oth
J´´anos Pach
BOLYAISOCIETY Geometry –
MATHEMATICALSTUDIES,24 Intuitive, Discrete, and Convex
pp.9–22.
List of Publications of La´szlo´´ Fejes To´th
Until 1946, L´aszl´oFejes T´oth published under the name of L´aszl´o Fejes.
1935
Dess´eriesexponentiellesdeCauchy.C.R.Acad.Sci.,Paris200(1935),1712–1714.
JFM 62.1191.03
1937
Soksz¨ogekre vonatkozo´´ sz´els˝˝o´´ert´ek feladatokr´ol. (Hungarian) [On extremum prob-
lems concerning polygons.] K¨oz´´episkolai Matematikai ´es Fizikai Lapok 13
(1937), 1–4.
1938
A Cauchy-fff´´ele exponencia´lis sor. (Hungarian) [The exponential series of Cauchy.]
Mat. Fiz. Lapok 45 (1938), 115–132. JFM 64.0284.04
Poli´´ederekre vonatkoz´´o sz´els˝˝o´´ert´ekfeladatok. (Hungarian) [Extremum problems
concerning polyhedra.] Mat. Fiz. Lapok 45 (1938), 191–199. JFM 64.0732.02
1939
U¨berdieApproximationkonvexerKurvendurchPolygonfolgen.CompositioMath.,
Groningen 6 (1939), 456–467. JFM 65.0822.03
Two inequalities concerning trigonometric polynomials. J. London Math. Soc. 14
(1939), 44–46. JFM 65.0254.01
U¨ber zwei Maximumaufgaben bei Polyedern. Tˆohoku Math. J. 46 (1939), 79–83.
MR0002194
Asimul´on-lapr´ol.(Hungarian)[Ontheapproximatingn-hedron.]Mat. Fiz. Lapok
46 (1939), 141–145. JFM 65.0827.01
10
ListofPublicationsofL´aszlo´´FejesT´´oth
1940
U¨ber einen geometrischen Satz. Math. Z. 46 (1940), 83–85. MR0001587
EineBemerkungzurApproximationdurchn-Eckringe.CompositioMath.7(1940),
474–476. MR0001588
Sur un th´´eor``eme concernant l’approximation des courbes par des suites de poly-
gones. Ann. Scuola Norm. Super. Pisa (2) 9 (1940), 143–145. MR0004993
Egyextrem´alissoklapr´ol.(Hungarian)[Onanextremalpolyhedron.]Math. Natur-
wiss. Anz. Ungar. Akad. Wiss. 59 (1940), 476–479. MR0015831
1942
Az egyenl˝ooldalu´´ h´´aromsz¨ogr´´acs, mint sz´´els˝˝o´´ert´ekfeladatok megold´asa. (Hungar-
ian) [The regular triangular lattice as the solution of extremum problems.]
Mat. Fiz. Lapok 49 (1942), 238–248. MR0017924
Aszab´a´lyostestek,mintsz´´els˝˝o´´ert´ekfeladatokmegold´asai.(Hungarian)[Theregular
polyhedraasthesolutionofextremumproblems.]Mat. Termeszett. E´rtes.61
(1942), 471–477. Zbl 0028.07604
Alehu˝˝l´´esFourier-sor´´ar´ol.(Hungarian)[OntheFourierseriesofthecooling.]Math.-
naturw. Anz. Ungar. Akad. Wiss. 61 (1942), 478–495. JFM 68.0144.03
1943
Einige Extremaleigenschaften des Kreisbogens bezu¨glich der Ann¨aherung durch
Polygone. Acta Sci. Math., Szeged 10 (1943), 164–173. Zbl 0028.09201
U¨ber eine Extremaleigenschaft der Kegelschnittbogen. Monatsh. Math. Phys. 50
(1943), 317–326. MR0010420
U¨ber die dichteste Kugellagerung. Math. Z. 48 (1943), 676–684. MR0009129
U¨ber eine Absch¨¨atzung des ku¨¨rzesten Abstandes zweier Punkte eines auf einer
Kugel߬acheliegendenPunktsystems.Jber. Deutsch. Math. Verein.53(1943),
66–68. MR0017539
Egy g¨ombfelu¨¨let befed´ese egybev´´ag´´o g¨¨ombsu¨vegekkel. (Hungarian) [Covering the
spherewithcongruentcaps.]Mat.Fiz.Lapok 50(1943),40–46.Zbl0060.34808
Az ellipszis izoperimetrikus ´es az ellipszoid izepif´an tulajdons´´aga´r´ol. (Hungarian)
[On the isoperimertric property of the ellipse and the isepiphane property of
the ellipsoid.] Math.-Naturw. Anz. Ungar. Akad. Wiss. 62 (1943), 88–94. Zbl
0061.38207
A g¨¨ombfelu¨¨letet egyenl˝o felsz´´nu˝˝ konvex r´´eszekre oszt´´o legr¨ovidebb g¨orbeh´´al´ozat.
(Hungarian) [The shortest net dividing the sphere into convex parts of
equal area.] Math. Naturwiss. Anz. Ungar. Akad. Wiss. 62 (1943), 349–354.
MR0024155