Hyperplane Arrangements with Large Average Diameter

The largest possible average diameter of a bounded cell of a simple hyperplane arrangement is conjectured to be not greater than the dimension. We prove that this conjecture holds in dimension 2, and is asymptotically tight in fixed dimension. We give the exact value of the largest possible average diameter for all simple arrangements in dimension 2, for arrangements having at most the dimension plus 2 hyperplanes, and for arrangements having 6 hyperplanes in dimension 3. In dimension 3, we give lower and upper bounds which are both asymptotically equal to the dimension.

[1]  Michael Shub,et al.  On the Curvature of the Central Path of Linear Programming Theory , 2003, Found. Comput. Math..

[2]  David Forge,et al.  On Counting the k-face Cells of Cyclic Arrangements , 2001, Eur. J. Comb..

[3]  Denis Naddef,et al.  The hirsch conjecture is true for (0, 1)-polytopes , 1989, Math. Program..

[4]  David Avis,et al.  Reverse Search for Enumeration , 1996, Discret. Appl. Math..

[5]  Klaus Meer.  Jean-Pierre Dedieu, Gregorio Malajovich, Mike Shub:: On the curvature of the central path of linear programming theory , 2006 .

[6]  V. Klee.  Paths on Polyhedra. I , 1965 .

[7]  C. Roos,et al.  Interior Point Methods for Linear Optimization , 2005 .

[8]  Gert Vegter,et al.  In handbook of discrete and computational geometry , 1997 .

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

[10]  R. W. Shannon.  Simplicial cells in arrangements of hyperplanes , 1979 .

[11]  Günter M. Ziegler,et al.  Higher bruhat orders and cyclic hyperplane arrangements , 1993 .

[12]  G. Ziegler.  Lectures on Polytopes , 1994 .

[13]  P. Lockhart.  INTRODUCTION TO GEOMETRY , 2007 .

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

[15]  Komei Fukuda,et al.  Complete combinatorial generation of small point configurations and hyperplane arrangements , 2001, CCCG.

[16]  Komei Fukuda,et al.  From the zonotope construction to the Minkowski addition of convex polytopes , 2004, J. Symb. Comput..

[17]  M. Spivak.  A comprehensive introduction to differential geometry , 1979 .

[18]  James Renegar,et al.  A mathematical view of interior-point methods in convex optimization , 2001, MPS-SIAM series on optimization.

[19]  Nesa L'abbe Wu,et al.  Linear programming and extensions , 1981 .

[20]  G. C. Shephard,et al.  Convex Polytopes , 1969, The Mathematical Gazette.

[21]  Komei Fukuda,et al.  Generation of Oriented Matroids—A Graph Theoretical Approach , 2002, Discret. Comput. Geom..

[22]  James G. Oxley,et al.  Matroid theory , 1992 .

[23]  Xun Dong.  The bounded complex of a uniform affine oriented matroid is a ball , 2008, J. Comb. Theory, Ser. A.

[24]  Nora Sleumer,et al.  Output-Sensitive Cell Enumeration in Hyperplane Arrangements , 1998, Nord. J. Comput..

[25]  Victor Klee,et al.  Many Polytopes Meeting the Conjectured Hirsch Bound , 1998, Discret. Comput. Geom..

[26]  Tamás Terlaky,et al.  Polytopes and arrangements: Diameter and curvature , 2008, Oper. Res. Lett..

[27]  Herbert Edelsbrunner,et al.  Algorithms in Combinatorial Geometry , 1987, EATCS Monographs in Theoretical Computer Science.

[28]  Godfried T. Toussaint,et al.  On Envelopes of Arrangements of Lines , 1996, J. Algorithms.

[29]  Bernd Sturmfels,et al.  On the coordinatization of oriented matroids , 1986, Discret. Comput. Geom..

[30]  K. Borgwardt.  The Simplex Method: A Probabilistic Analysis , 1986 .

[31]  James R. Munkres,et al.  Elements of algebraic topology , 1984 .

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

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

[35]  Yan-Bin Jia,et al.  The simplex method , 2019, 100 Years of Math Milestones.

[36]  Franz Aurenhammer.  Using Gale Transforms in Computational Geometry , 1988, Workshop on Computational Geometry.

[37]  V. Klee,et al.  Thed-step conjecture for polyhedra of dimensiond<6 , 1967 .

[38]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[39]  Tamás Terlaky,et al.  A Continuous d-Step Conjecture for Polytopes , 2009, Discret. Comput. Geom..

[40]  Akihisa Tamura,et al.  Combinatorial face enumeration in arrangements and oriented matroids , 1991, Discret. Appl. Math..

[41]  Juergen Bokowski.  Computational Oriented Matroids , 2006 .

[42]  中山 裕貴,et al.  Methods for realizations of oriented matroids and characteristic oriented matroids , 2007 .