Current Open Problems in Discrete and Computational Geometry

We have selected problems that may not yet be well known, but have the potential to push the research in interesting directions. In particular, we state problems that do not require specific knowledge outside the standard circle of ideas in discrete geometry. Despite the relatively simple statements, these problems are related to current research and their solutions are likely to require new ideas and approaches. We have chosen problems from different fields to make this short paper attractive to a wide range of specialists. The article is published in the author’s wording.

[1]  Haim Kaplan,et al.  Simple Proofs of Classical Theorems in Discrete Geometry via the Guth–Katz Polynomial Partitioning Technique , 2011, Discret. Comput. Geom..

[2]  Herbert Edelsbrunner,et al.  Computational Topology - an Introduction , 2009 .

[3]  Toshinori Sakai,et al.  Balanced Convex Partitions of Measures in ℝ2 , 2002, Graphs Comb..

[4]  M. Gromov.  Singularities, Expanders and Topology of Maps. Part 2: from Combinatorics to Topology Via Algebraic Isoperimetry , 2010 .

[5]  P. Borodin.  An example of nonexistence of a steiner point in a Banach space , 2010 .

[6]  V. A. Vasil.  BRAID GROUP COHOMOLOGIES AND ALGORITHM COMPLEXITY , 2006 .

[7]  A. Harnack,et al.  Ueber die Vieltheiligkeit der ebenen algebraischen Curven , 1876 .

[8]  Alexander Russell,et al.  Computational topology: ambient isotopic approximation of 2-manifolds , 2003, Theor. Comput. Sci..

[9]  I. Bárány,et al.  Equipartitioning by a convex 3-fan , 2010 .

[10]  A. Ivanov,et al.  Geometry of minimal networks and the one-dimensional Plateau problem , 1992 .

[11]  Thomas Auf der Heyde,et al.  On Quantifying Chirality , 1992 .

[12]  R. Adler.  The Geometry of Random Fields , 2009 .

[13]  B. M. Fulk.  MATH , 1992 .

[14]  Z. Füredi,et al.  The number of triangles covering the center of an n-set , 1984 .

[15]  V. Kadets.  Under a suitable renorming every nonreflexive Banach space has a finite subsetwithout a Steiner point , 2013 .

[16]  Adrian L. Melott,et al.  Topology of the large-scale structure of the universe , 1990 .

[17]  János Pach.  A Tverberg-type result on multicolored simplices , 1998, Comput. Geom..

[18]  Uniqueness of Steiner minimal trees on boundaries in general position , 2006 .

[19]  A. Ivanov,et al.  One-dimensional Gromov minimal filling problem , 2012 .

[20]  Konrad J. Swanepoel.  The local Steiner problem in normed planes , 2000, Networks.

[21]  B. Aronov,et al.  Convex Equipartitions of volume and surface area , 2010, 1010.4611.

[22]  Libor Veselý,et al.  A characterization of reflexivity in the terms of the existence of generalized centers , 1993 .

[23]  R. Adler,et al.  The Geometry of Random Fields , 1982 .

[24]  Ming-Kuei Hu,et al.  Visual pattern recognition by moment invariants , 1962, IRE Trans. Inf. Theory.

[25]  R. Karasev.  Equipartition of several measures , 2010, 1011.4762.

[26]  Pablo Soberón,et al.  BALANCED CONVEX PARTITIONS OF MEASURES IN ℝ d , 2010, 1010.6191.

[27]  N. Innami,et al.  A comparison theorem for Steiner minimum trees in surfaces with curvature bounded below , 2013 .

[28]  A. Ivanov,et al.  Branching geodesics in normed spaces , 2002 .

[29]  Imre Bárány,et al.  A generalization of carathéodory's theorem , 1982, Discret. Math..

[30]  J. Tukey,et al.  Generalized “sandwich” theorems , 1942 .

[31]  David Eppstein,et al.  Worst-case bounds for subadditive geometric graphs , 1993, SCG '93.

[32]  A. Ivanov,et al.  Branching solutions to one-dimensional variational problems , 2001 .

[33]  A. Buda,et al.  On a measure of axiality for triangular domains. , 1991 .

[34]  K. I. Oblakov.  Non-existence of distinct codirected locally minimal trees on a plane , 2009 .

[35]  B. Grünbaum.  Measures of symmetry for convex sets , 1963 .

[36]  H. Steinhaus,et al.  Sur la division des ensembles de l'espace par les plans et des ensembles plans par les cercles , 1945 .

[37]  A.O.Ivanov,et al.  One-dimensional Gromov minimal filling , 2010, 1101.0106.

[38]  Roman N. Karasev,et al.  A Simpler Proof of the Boros–Füredi–Bárány–Pach–Gromov Theorem , 2010, Discret. Comput. Geom..

[39]  Jirí Matousek,et al.  On Gromov’s Method of Selecting Heavily Covered Points , 2011, Discret. Comput. Geom..