Realization Spaces of Arrangements of Convex Bodies

We introduce combinatorial types of arrangements of convex bodies, extending order types of point sets to arrangements of convex bodies, and study their realization spaces. Our main results witness a trade-off between the combinatorial complexity of the bodies and the topological complexity of their realization space. First, we show that every combinatorial type is realizable and its realization space is contractible under mild assumptions. Second, we prove a universality theorem that says the restriction of the realization space to arrangements polygons with a bounded number of vertices can have the homotopy type of any primary semialgebraic set.

[1]  Andreas F. Holmsen,et al.  Interval sequences and the combinatorial encoding of planar families of pairwise disjoint convex sets , 2005 .

[2]  G. Ringel.  Teilungen der Ebene durch Geraden oder topologische Geraden , 1956 .

[3]  Harmonics Book,et al.  Geometric Applications Of Fourier Series And Spherical Harmonics , 2016 .

[4]  Alfredo Hubard,et al.  Order Types of Convex Bodies , 2010, Order.

[5]  Jürgen Richter-Gebert.  The Universality Theorems for Oriented Matroids and Polytopes , 1999 .

[6]  Alfredo Hubard,et al.  THE ERDŐS–SZEKERES PROBLEM FOR NON-CROSSING CONVEX SETS , 2013, 1305.2266.

[7]  Peter W. Shor,et al.  Stretchability of Pseudolines is NP-Hard , 1990, Applied Geometry And Discrete Mathematics.

[8]  János Pach,et al.  Families of convex sets not representable by points , 2008 .

[9]  Jacob E. Goodman,et al.  Proof of a conjecture of Burr, Grünbaum, and Sloane , 1980, Discret. Math..

[10]  Richard Pollack,et al.  Upper bounds for configurations and polytopes inRd , 1986, Discret. Comput. Geom..

[11]  Herbert Edelsbrunner,et al.  Semispaces of Configurations , 1987 .

[12]  Richard Pollack,et al.  Semispaces of Configurations, Cell Complexes of Arrangements , 1984, J. Comb. Theory, Ser. A.

[13]  Stefan Felsner,et al.  Coding and Counting Arrangements of Pseudolines , 2011, Discret. Comput. Geom..

[14]  Jim Lawrence,et al.  Oriented matroids , 1978, J. Comb. Theory B.

[15]  Michel Pocchiola,et al.  Arrangements of double pseudolines , 2011, ArXiv.

[16]  Richard Pollack,et al.  On the Combinatorial Classification of Nondegenerate Configurations in the Plane , 1980, J. Comb. Theory, Ser. A.

[17]  M. Kapovich,et al.  Universality theorems for configuration spaces of planar linkages , 1998, math/9803150.

[18]  Yasuyuki Tsukamoto.  New Examples of Oriented Matroids with Disconnected Realization Spaces , 2013, Discret. Comput. Geom..

[19]  B. Grünbaum.  Arrangements and Spreads , 1972 .

[20]  Andreas F. Holmsen,et al.  Regular systems of paths and families of convex sets in convex position , 2013, 1307.0909.

[21]  Mordechai Novick.  Allowable Interval Sequences and Line Transversals in the Plane , 2012, Discret. Comput. Geom..

[22]  Jürgen Richter-Gebert.  Realization Spaces of Polytopes , 1996 .

[23]  Ravi Vakil.  Murphy’s law in algebraic geometry: Badly-behaved deformation spaces , 2004 .

[24]  N. Mnev.  The universality theorems on the classification problem of configuration varieties and convex polytopes varieties , 1988 .

[25]  B. Sturmfels.  Oriented Matroids , 1993 .

[26]  P. Gritzmann,et al.  Applied geometry and discrete mathematics : the Victor Klee festschrift , 1991 .

[27]  Mordechai Novick.  Allowable Interval Sequences and Separating Convex Sets in the Plane , 2012, Discret. Comput. Geom..

[28]  Jürgen Richter-Gebert,et al.  Mnev's Universality Theorem revisited , 1995 .

[29]  Louis Theran,et al.  Delaunay triangulations with disconnected realization spaces , 2014, SoCG.

[30]  Donald E. Knuth,et al.  Axioms and Hulls , 1992, Lecture Notes in Computer Science.

[31]  Richard Pollack,et al.  The combinatorial encoding of disjoint convex sets in the plane , 2008, Comb..