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Mathematisches Forschungsinstitut Oberwolfach Discrete Geometry Mathematisches Forschungsinstitut Oberwolfach Report No. 44/2011 DOI: 10.4171/OWR/2011/44 Discrete Geometry Organised by Jiˇr´ıMatouˇsek (Praha) G¨unter Rote (Berlin) Imre B´ar´any (Budapest, London) September 4th – September 10th, 2011 Abstract. A number of remarkable recent developments in many branches of discrete geometry have been presented at the workshop, some of them demonstrating strong interactions with other fields of mathematics (such as algebraic geometry, harmonic analysis, and topology). The field is very active with lots of open questions and many solutions. There was a large number of young participants who are eager to work on these problems. The future of discrete geometry looks more than promising. Mathematics Subject Classification (2000): 52Cxx. Introduction by the Organisers Discrete Geometry deals with the structure and complexity of discrete geometric objects ranging from finite point sets in the plane to more complex structures like arrangements of n-dimensional convex bodies. Classical problems such as Kepler’s conjecture and Hilbert’s third problem on decomposing polyhedra, as well as clas- sical works by mathematicians such as Minkowski, Steinitz, Hadwiger and Erd˝os are part of the heritage of this area. In the last couple of years several outstanding open problems have been solved. Here we list a few of them: (1) Erd˝os distinct distances problem by Guth and Katz, using algebraic geometry (based on ideas of Elekes and Sharir), (2) tight lower bounds for geometric ε-nets by Pach and Tar- dos, and a weaker but still superlinear lower bound by Alon, (3) a superlinear lower bound on the size of weak ε-nets by Bukh, Matouˇsek, and Nivasch, (4) disproof by Santos of the famous Hirsch conjecture from 1957, (5) topological extension of the first selection lemma by Gromov (which also improves the constant) and which has lead to further use of algebraic topology in discrete geometry. 2460 Oberwolfach Report 44/2011 By its nature, this area is interdisciplinary and has relations to many other vital mathematical fields. The breakthrough results above use methods of algebraic ge- ometry, topology, combinatorics, computational geometry, convexity, discrepancy theory, and probability. At the same time it is on the cutting edge of applica- tions such as geographic information systems, mathematical programming, coding theory, solid modelling, computational structural biology and crystallography. The workshop was attended by 47 participants. There was a series of 10 survey talks giving an overview of developments in Discrete Geometry and related fields: Micha Sharir: From Joints to Distinct Distances and Beyond: The Dawn • of an Algebraic Era in Combinatorial Geometry Roman Karasev: A simpler proof of the Boros–F¨uredi–B´ar´any–Pach–Gro- • mov theorem J´anos Pach: Piercing convex sets • Luis Montejano: When is a disk trapped with four lines? • Uli Wagner: Isoperimetry, Crossing Numbers, and Multiplicities of (Equi- • variant) Maps J´ozsef Solymosi: Point-pseudoline incidences in higher dimensions • Boris Bukh: Space crossing numbers • Nati Linial: What are high-dimensional permutations? How many are • there? Igor Pak: Finite tilings • G¨unter M. Ziegler: Polytopes with low-dimensional realization spaces • In addition, there were 26 shorter talks and an open problem session chaired by J´anos Pach on Tuesday evening—a collection of open problems resulting from this session can be found in this report. The program left ample time for research and discussions in the stimulating atmosphere of the Oberwolfach Institute. In particular, there were several special informal sessions, attanded by smaller groups of the participants, on specific topics of common interest. On Wednesday we had a very pleasant excursion leading to the MiMa (Museum for Minerals and Mathematics in Oberwolfach Kirche) where many participants of the workshop (with the help of the museum’s staff) worked on the construction of the “Exploded stellated Dodecahedron” using Zometool building blocks. Discrete Geometry 2461 Workshop: Discrete Geometry Table of Contents Micha Sharir From Joints to Distinct Distances and Beyond: The Dawn of an Algebraic Era in Combinatorial Geometry ..........................2465 Roman Karasev A simpler proof of the Boros–F¨uredi–B´ar´any–Pach–Gromov theorem ...2469 George Purdy When is the number of hyperplanes determined by n points in d-space at least the number of (d 2)-dimensional flats? .......................2472 − Ben Lund A pseudoline counterexample to the Strong Dirac conjecture ...........2474 Xavier Goaoc (joint with Eric´ Colin de Verdi`ere and Gr´egory Ginot) Helly numbers of acyclic families ..................................2477 Otfried Cheong (joint with Antoine Vigneron) Generalizations of the Kakeya problem .............................2479 Martin Tancer (joint with Dmitry Tonkonog) Good covers are algorithmically unrecognizable ......................2481 Pavel Valtr Equalities on empty polygons ......................................2482 J´anos Pach (joint with G´abor Tardos) Piercing quasi-rectangles — On a problem of Danzer and Rogers ......2483 Luis Montejano (joint with Tudor Zamfirescu) When is a disk trapped with four lines? ............................2485 Matthias Beck (joint with Benjamin Braun, Ira Gessel, Nguyen Le, Sunyoung Lee, Carla Savage) Integer partitions from a geometric viewpoint .......................2488 Andr´as Bezdek On stability of polyhedra ..........................................2490 Anders Bj¨orner A cell complex in number theory ...................................2492 Zolt´an F¨uredi (joint with Mikl´os Ruszink´o) Uniform hypergraphs containing no grids, a problem concerning superimposed codes ..............................................2492 2462 Oberwolfach Report 44/2011 Martin Henk (joint with Iskander Aliev, Lenny Fukshansky) Various aspects of Frobenius numbers ..............................2495 Hiroshi Maehara To hold a convex body by a circle ..................................2498 Uli Wagner Isoperimetry, Crossing Numbers, and Multiplicities of (Equivariant) Maps ................................................... .......2500 J´ozsef Solymosi (joint with Terence Tao) Point-pseudoline incidences in higher dimensions ....................2502 Stefan Langerman (joint with Vida Dujmovi´c) A center transversal theorem for hyperplanes ........................2503 Josef Cibulka (joint with Jan Kynˇcl) Tight bounds on the maximum size of a set of permutations of bounded VC-dimension ..................................................2503 Boris Bukh (joint with Alfredo Hubard) Space crossing numbers ..........................................2506 Nati Linial (joint with Zur Luria) What are high-dimensional permutations? How many are there? ......2508 Rom Pinchasi On a problem of Gr¨unbaum and Motzkin, and Erd˝os and Purdy .......2511 D¨om¨ot¨or P´alv¨olgyi (joint with J´anos Pach, G´eza T´oth) New results on decomposability of geometric coverings ................2511 Pablo Sober´on (joint with Ricardo Strausz) On the tolerated Tverberg Theorem ................................2515 Edgardo Rold´an-Pensado (joint with Jes´us Jer´onimo-Castro) On line transversals .............................................2516 Norihide Tokushige (joint with Hidehiko Kamiya, Akimichi Takemura) Counting the number of ranking patterns ...........................2518 G´abor Tardos Construction of locally plane graphs ................................2519 Igor Pak (joint with Jed Yang) Finite tilings ................................................... 2522 G¨unter M. Ziegler (joint with Karim Adiprasito) Polytopes with low-dimensional realization spaces ....................2522 Csaba D. T´oth (joint with Adrian Dumitrescu) Anchored rectangle packing .......................................2525 Discrete Geometry 2463 Frank Vallentin (joint with Fernando Mario de Oliveira Filho) Upper bounds for densest packings with congruent copies of a convex body ................................................... ........2527 Vladimir Dolnikov (joint with Grigory Chelnokov) On transversals of quasialgebraic families of sets ....................2530 Arseniy Akopyan (joint with Roman Karasev) Kadets type theorems for partitions of a convex body .................2532 Kenneth L. Clarkson Remark on Coresets for Minimum Enclosing Ellipsoids ...............2534 Konrad J. Swanepoel Dense favourite-distance digraphs ..................................2535 Open problems in Discrete Geometry ...............................2538 Discrete Geometry 2465 Abstracts From Joints to Distinct Distances and Beyond: The Dawn of an Algebraic Era in Combinatorial Geometry Micha Sharir Summary. In the past three years the landscape of combinatorial geometry has considerably changed, due to two groundbreaking papers by Guth and Katz ([6] in 2008 and [7] in 2010). They have introduced reasonably simple techniques from algebraic geometry that enabled them to tackle successfully several major problems in combinatorial geometry. Their first paper obtained a complete solution to the joints problem, a problem involving incidences between points and lines in three dimensions which has been open since it was first posed (by myself and others) in 1992. The second Guth–Katz paper was even more dramatic. They obtained a nearly complete solution to the classical problem of Erd˝os [4] on distinct distances in the plane, which was open since 1946. Both problems have been extensively studied over the years, using more traditional, and progressively more complex methods
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