Discrete and Computational Geometry: The Goodman-Pollack Festschrift

846 Pages·2003·21.92 MB·English

by  Pankaj K. Agarwal

Algorithms and Combinatorics 25
Editorial Board
R.L. Graham, La Jolla B. Korte, Bonn
L. Lova´sz, Budapest A. Wigderson, Princeton
G.M. Ziegler, Berlin
Springer-Verlag Berlin Heidelberg GmbH

Boris Aronov • Saugata Basu
Ja´nos Pach • Micha Sharir
Editors
Discrete and Computational
Geometry
The Goodman-Pollack Festschrift
With287Figures
123
Editors
BorisAronov Ja´nosPach
DepartmentofComputer CityCollege
andInformationScience andCourantInstitute
PolytechnicUniversity 251MercerSt.
SixMetroTechCenter NewYork,NY10012,USA
Brooklyn,NY11201,USA e-mail:[email protected]
e-mail:[email protected]
MichaSharir
SaugataBasu SchoolofComputerScience
SchoolofMathematics TelAvivUniversity
GeorgiaInstituteofTechnology TelAviv69978,Israel
Atlanta,GA30332-0160,USA e-mail:[email protected]
e-mail:[email protected]
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MathematicsSubjectClassification(2000):
52-XX,68-XX,05-XX,14-XX,57-XX,60-02
ISSN0937-5511
ISBN 978-3-642-62442-1 ISBN 978-3-642-55566-4 (eBook)
DOI 10.1007/978-3-642-55566-4
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Preface
Eli Goodman and Ricky Pollack were born at about the same time and
brought up in the same city, within the same social milieu. They had many
musical, literary, and scientific interests in common that drew them to the
same events and locations. Their orbits were bound to intersect.
Preciselywhereandwhenithappened,theydisagree. Wasitataconcert
in the late fifties? In a group of bridge enthusiasts? Or perhaps a little
later, at the Heights campus of New York University, where each of them
taught? In fact, Eli and Ricky disagree about a greatmany little things and
details: thisseemstobetheirworkingmethod,thesecretoftheirrelationship
– carefully testing commonly accepted views and concepts, no matter how
self-evidenttheymightappear,byexposingthemtoeachother’scriticaleyes
and words.
Bothofthemreceivedafirst-classmathematicaleducation. Ricky’sthesis
was in number theory,under the supervisionof HaroldShapiro atNew York
University, while Eli became an algebraic geometer under the direction of
HeisukeHironakaandStevenKleimanatColumbiaUniversity. Bothofthem
publishedremarkableresultsintheirrespectiveresearchareasandhadataste
of scientific success. In the mid 1970s, apparently on divergent paths, both
of them took sabbaticals,and soon after eachdecided to try something new.
RickyspentasemesteratMcGillUniversity,collaboratingwithagroupof
excellent combinatorists. In particular, he learned about and was fascinated
bytheErdo˝s–Szekeresconjecture,whichstatesthatineverysetofmorethan
2n−2pointsingeneralpositionintheplaneonecanfindnpointsthatformthe
vertexset ofa convexn-gon. He spent two months trying to come up with a
proof. Unbeknownsttohim,Elihadbeenworkingintopologicalgraphtheory
and had recently turned his attention to some problems in combinatorial
geometry. Shortly after Ricky’s return from Montreal, he bumped into Eli
at Courant Institute, at a seminar talk on graphalgorithms. “What are you
doinghere?” –theyaskedeachotherinsurprise. AfewweekslaterElicalled
Ricky: – “Have you ever heard of the Erd˝os–Szekeresproblem? I have some
ideas ...”
VI Preface
They started meeting once or twice a week and, based on Eli’s prelimi-
naryideas,theysoondevelopedanaturalcombinatorialencodingofaplanar
point configuration. This later became known as an “allowable sequence,”
and turned out to be an important tool in combinatorial geometry that led
to spectacular breakthroughs such as Ungar’s “book” proof of Scott’s con-
jecture, according to which every set of n noncollinear points in the plane
determinesatleast2(cid:2)n/2(cid:3)distinctdirections. Dozensofjointpapersevolved
from this first fruitful collaboration,some of them developing theoretical as-
pects of the material, others solving problems that had seemed insurmount-
able by other methods. To their surprise, Eli and Ricky found themselves
drawn more to combinatorial geometry than to their original fields. They
loved this newly-found playground, and made it their own. They made a
permanent move and never looked back. Gradually their joint work spread
out to encompass various related topics in discrete geometry, and later in-
cludedadditionalauthorsaswell. Oftheirmanyground-breakingdiscoveries
we mention two. (1) They revitalized geometric transversaltheory by show-
ing that, using the notion of “order types,” one can establish many exciting
Helly-type theorems in d-space for the existence of a k-dimensional flat in-
tersecting all members of a family of convex bodies, for 0<k <d. (2) They
were among the first mathematicians to use real algebraic geometry to enu-
merate order types, simplicial polytopes, andother geometricobjects. Their
surprising results found numerous applications in ray shooting, in computer
graphics, and elsewhere.
In 1984, after it had become clear that their work had implications not
onlyinthewell-establisheddisciplineofdiscretegeometrybutalsointhethen
nascent field of computational geometry, the opportunity arose of founding
a journal devoted to both fields. Three publishers were interested, but Eli
and Ricky decided on Springer-Verlag (whose New York office was then un-
der the leadership of Walter Kaufmann-Bu¨hler) as the best of the choices,
and the journal Discrete & Computational Geometry was born the follow-
ing year. Its first issue appeared in 1986, and the journal has thrived since
then, having grown from under 400 pages a year at the start to nearly 1300
pages currently. At the same time the field has grown enormously. It is
hard to overestimate the role Eli, Ricky, and their journal have played in
these developments. The marriageof the two fields provedto be fruitful and
long-lasting. DCG is now the leading journal in a fast growing area along
the borderline of pure mathematics and computer science. Eli and Ricky
co-organizedthe first Special Year at DIMACS (1989–90)and severalmajor
conferencesdedicatedtothesubject(SantaCruz1986,MountHolyoke1996,
Ascona 1999,and Oberwolfach2000),as well as a number of specialsessions
atmeetings ofthe AmericanMathematicalSociety. The investmentofseem-
ingly infinite resources on their part has paid off: they have built bridges
betweenremote groups ofresearchersworkingon similarproblems as wellas
bridges between various subdisciplines. In this sense, they are perhaps the
Preface VII
two most conspicuous “founding fathers” of the field. As such, their names
have become inseparable.
The idea of editing a collection of papers dedicated to Eli and Ricky
was proposed by N. Prabhu, a former student of Ricky’s, to whom we are
grateful. Weweresoonoverwhelmedbyanavalancheofexcellentmanuscripts
submitted by leading researchers in discrete and computational geometry.
They contributed state-of-the-art results covering virtually every aspect of
the field. In the selection process, we followed the simple rule that each
paper should be refereed according to the high standards adopted by Eli’s
andRicky’s journal. It is mostappropriatethat the publisher ofthis volume
is Springer-Verlag, under whose auspices the journal is produced. We are
very grateful for their enthusiastic support.
Each editor of this volume regards himself as a student, a close friend, a
collaborator,andaco-authorofEliandRicky–allinone. Theyhaveshaped
our lives, our mathematical thinking, our views of the world. We wish that
their golden age last forever to the benefit of our community.
Boris Aronov
Saugata Basu
J´anos Pach
Micha Sharir
Contents
On the Complexity of Many Faces in Arrangements of Pseudo-Segments
and of Circles ......................................................... 1
P.K. Agarwal, B. Aronov and M. Sharir
PolyhedralCones of Magic Cubes and Squares ........................ 25
M. Ahmed, J. De Loera and R. Hemmecke
Congruent Dudeney Dissections of Triangles and Convex Quadrilaterals
– All Hinge Points Interior to the Sides of the Polygons ............... 43
J. Akiyama and G. Nakamura
Computing the Hausdorff Distance of Geometric Patterns and Shapes 65
H. Alt, P. Braß, M. Godau, C. Knauer and C. Wenk
A Sum of Squares Theorem for Visibility Complexes and Applications 77
P. Angelier and M. Pocchiola
On the Reflexivity of Point Sets ....................................... 139
E.M. Arkin, S.P. Fekete, F. Hurtado, J.S.B. Mitchell, M. Noy,
V. Sacrista´n and S. Sethia
Geometric Permutations of Large Families of Translates ............... 157
A. Asinowski, A. Holmsen, M. Katchalski and H. Tverberg
Integer Points in Rotating Convex Bodies ............................. 177
I. Ba´ra´ny and J. Matouˇsek
Complex Matroids – Phirotopes and Their Realizations in Rank 2 ..... 203
A. Below, V. Krummeck and J. Richter-Gebert
Covering the Sphere by Equal Spherical Balls ......................... 235
K. Bo¨ro¨czky, Jr. and G. Wintsche
X Contents
Lower Bounds for High Dimensional Nearest Neighbor Search
and Related Problems ................................................ 253
A. Borodin, R. Ostrovsky and Y. Rabani
A Tura´n-type Extremal Theory of Convex Geometric Graphs ......... 275
P. Brass, G. Ka´rolyi and P. Valtr
On the Inapproximability of Polynomial-programming,the Geometry
of Stable Sets, and the Power of Relaxation ........................... 301
A. Brieden and P. Gritzmann
A Lower Bound on the Complexity of Approximate Nearest-Neighbor
Searching on the Hamming Cube ..................................... 313
A. Chakrabarti, B. Chazelle, B. Gum and A. Lvov
Detecting Undersampling in Surface Reconstruction ................... 329
T.K. Dey and J. Giesen
A Survey of the Hadwiger-Debrunner (p,q)-problem ................... 347
J. Eckhoff
Surface Reconstruction by Wrapping Finite Sets in Space ............. 379
H. Edelsbrunner
Infeasibility of Systems of Halfspaces .................................. 405
S. Felsner and N. Morawe
Complete CombinatorialGeneration of Small Point Configurations
and Hyperplane Arrangements ........................................ 425
L. Finschi and K. Fukuda
Relative Closure and the Complexity of Pfaffian Elimination .......... 441
A. Gabrielov
Are Your Polyhedra the Same as My Polyhedra? ...................... 461
B. Gru¨nbaum
Some Algorithms Arising in the Proof of the Kepler Conjecture ....... 489
T.C. Hales
The Minimal Number of Triangles Needed to Span a Polygon
Embedded in Rd ...................................................... 509
J. Hass and J.C. Lagarias
Jacobi Decomposition and Eigenvalues of Symmetric Matrices ......... 527
W. He and N. Prabhu
Contents XI
Discrete Geometry on Red and Blue Points in the Plane
– A Survey – ..........................................................551
A. Kaneko and M. Kano
Configurations with Rational Angles and Trigonometric Diophantine
Equations .............................................................571
M. Laczkovich
Reconstructing Sets From Interpoint Distances ........................ 597
P. Lemke, S. Skiena and W.D. Smith
Dense Packings of Congruent Circles in Rectangles with a Variable
Aspect Ratio ......................................................... 633
B.D. Lubachevsky and R. Graham
Colorings and Homomorphisms of Minor Closed Classes ............... 651
J. Neˇsetˇril and P. Ossona de Mendez
Conflict-free Colorings ................................................ 665
J. Pach and G. To´th
New Complexity Bounds for Cylindrical Decompositions
of Sub-Pfaffian Sets ................................................... 673
S. Pericleous and N. Vorobjov
Note on the Chromatic Number of the Space .......................... 695
R. Radoiˇci´c and G. To´th
Expansive Motions and the Polytope of Pointed Pseudo-Triangulations 699
G. Rote, F. Santos and I. Streinu
Some Recent Quantitative and Algorithmic Results in Real Algebraic
Geometry ............................................................. 737
M.-F. Roy
A Discrete Isoperimetric Inequality and Its Application
to Sphere Packings ....................................................751
P. Scholl, A. Schu¨rmann and J.M. Wills
On the Number of Maximal Regular Simplices Determined
by n Points in Rd ..................................................... 767
Z. Schur, M.A. Perles, H. Martini and Y.S. Kupitz
Balanced Lines, Halving Triangles, and the Generalized Lower Bound
Theorem .............................................................. 789
M. Sharir and E. Welzl