Geodesic
Current Open Problems in Discrete and Computational Geometry
Modelirovanie i analiz informacionnyh sistem, Tome 19 (2012) no. 5, pp. 5-17.

Voir la notice de l'article provenant de la source Math-Net.Ru

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.
Keywords: discrete and computational geometry, computational topology, open problems. 
@article{MAIS_2012_19_5_a0,
     author = {H. Edelsbrunner and A. Ivanov and R. Karasev},
     title = {Current {Open} {Problems} in {Discrete}  and {Computational} {Geometry}},
     journal = {Modelirovanie i analiz informacionnyh sistem},
     pages = {5--17},
     publisher = {mathdoc},
     volume = {19},
     number = {5},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/MAIS_2012_19_5_a0/}
}
TY  - JOUR
AU  - H. Edelsbrunner
AU  - A. Ivanov
AU  - R. Karasev
TI  - Current Open Problems in Discrete  and Computational Geometry
JO  - Modelirovanie i analiz informacionnyh sistem
PY  - 2012
SP  - 5
EP  - 17
VL  - 19
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MAIS_2012_19_5_a0/
LA  - en
ID  - MAIS_2012_19_5_a0
ER  - 
%0 Journal Article
%A H. Edelsbrunner
%A A. Ivanov
%A R. Karasev
%T Current Open Problems in Discrete  and Computational Geometry
%J Modelirovanie i analiz informacionnyh sistem
%D 2012
%P 5-17
%V 19
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MAIS_2012_19_5_a0/
%G en
%F MAIS_2012_19_5_a0
H. Edelsbrunner; A. Ivanov; R. Karasev. Current Open Problems in Discrete  and Computational Geometry. Modelirovanie i analiz informacionnyh sistem, Tome 19 (2012) no. 5, pp. 5-17. http://geodesic.mathdoc.fr/item/MAIS_2012_19_5_a0/

[1] R. L. Adler, The Geometry of Random Fields, John Wiley Sons, Chichester, England, 1981 | MR | Zbl

[2] H. Edelsbrunner, J. L. Harer, Computational Topology. An Introduction, Amer. Math. Soc., Providence, Rhode Island, 2010 | MR | Zbl

[3] A. J. S. Hamilton, J. R. Gott III, D. Weinberg, “The topology of the large-scale structure of the Universe”, The Astrophys. J., 309 (1986), 1–12 | DOI | MR

[4] A. B. Buda, T. Auf der Heyde, K. Mislow, “On quantifying chirality”, Angew. Chem., 31 (1992), 989–1007 | DOI

[5] A. B. Buda, K. Mislow, “On a measure of axiality for triangular domains”, Elem. Math., 46 (1999), 65–73 | MR

[6] B. Grünbaum, “Measures of symmetry for convex sets”, Proc. Sympos. Pure Math., 7, Amer. Math. Soc., 1963 | MR

[7] M. K. Hu, “Visual pattern recognition by moment invariants”, IEEE Trans. Inform. Theory, 8 (1962)

[8] B. Aronov, A. Hubard, Convex equipartitions of volume and surface area, 2010, arXiv: 1010.4611

[9] I. Bárány, “A generalization of Carathéodory's theorem”, Discrete Math., 40:2–3 (1982), 141–152 | DOI | MR

[10] I. Bárány, P. Blagojević, A. Szűcs, “Equipartitioning by a convex $3$-fan”, Adv. Math., 223:2 (2010), 579–593 | DOI | MR | Zbl

[11] M. Bern, D. Eppstein, “Worst-case bounds for subadditive geometric graphs”, Proc. 9th Ann. Sympos. Comput. Geom., 1993, 183–188

[12] E. Boros, Z. Füredi, “The number of triangles covering the center of an $n$-set”, Geom. Dedicata, 17:1 (1984), 69–77 | DOI | MR | Zbl

[13] J. Pach, “A Tverberg-type result on multicolored simplices”, Comput. Geom., 10:2 (1998), 71–76 | DOI | MR | Zbl

[14] C. G. A. Harnack, “Über Vieltheiligkeit der ebenen algebraischen Curven”, Math. Ann., 10 (1876), 189–199 | DOI | MR

[15] M. Gromov, “Singularities, expanders, and topology of maps. Part 2: From combinatorics to topology via algebraic isoperimetry”, Geometric and Functional Analysis, 20:2 (2010), 416–526 | DOI | MR | Zbl

[16] R. N. Karasev, Equipartition of several measures, 2010, arXiv: 1011.4762 | MR

[17] R. N. Karasev, “A simpler proof of the Boros–Füredi–Bárány–Pach–Gromov theorem”, Discrete Comput. Geom., 47:3 (2012), 492–495 | DOI | MR | Zbl

[18] H. Kaplan, J. Matoušek, M. Sharir, “Simple proofs of classical theorems in discrete geometry via the Guth–Katz polynomial partitioning technique”, Discrete Comput. Geom., 48:3 (2012), 499–517 | DOI | MR

[19] R. Nandakumar, N. Ramana Rao, Fair' partitions of polygons — an introduction, 2008, arXiv: 0812.2241

[20] J. Matoušek, U. Wagner, On Gromov's method of selecting heavily covered points, 2011, arXiv: 1102.3515

[21] P. Soberón, Balanced convex partitions of measures in $\mathbb{R}^d$, 2010, arXiv: 1010.6191 | MR

[22] H. Steinhaus, “Sur la division des ensembles de l'espaces par les plans et des ensembles plans par les cercles”, Fund. Math., 33 (1945), 245–263 | MR | Zbl

[23] A. H. Stone, J. W. Tukey, “Generalized 'sandwich' theorems”, Duke Math. J., 9 (1942), 356–359 | DOI | MR | Zbl

[24] Funct. Anal. Appl., 22:3 (1988), 182–190 | DOI | MR

[25] Russian Math. Surveys, 47:2 (1992), 59–131 | DOI | MR | Zbl

[26] N. Innami, S. Naya, “A comparison theorems for Steiner minimum trees in surfaces with curvature bounded below”, Tohoku Math. Journal, 2012 (to appear)

[27] A. O. Ivanov, A. A. Tuzhilin, Extreme Networks Theory, Inst. of Komp. Issl., Moscow–Izhevsk, 2003 (in Russian)

[28] L. Vesely, “A characterization of reflexivity in the terms of the existence of generalized centers”, Extracta Mathematicae, 8:2–3 (1993), 125–131 | MR | Zbl

[29] Math. Notes, 87:4 (2010), 485–488 | DOI | DOI | MR | Zbl

[30] B. B. Bednov, N. P. Strelkova, “On the existence problem for shortest networks in Banach spaces”, Mat. zametki, 2013 (to appear)

[31] V. Kadets, “Under a suitable renorming every nonreflexive Banach space has a finite subset without a Steiner point”, Matematychni Studii, 36:2 (2011), 197–200 | MR

[32] A. O. Ivanov and A. A. Tuzhilin, Branching Solutions to One-Dimensional Variational Problems, World Scientific, Singapore–New Jersey–London–Hong Kong, 2000 | MR

[33] A. O. Ivanov, A. A. Tuzhilin, “Branching geodesics in normed spaces”, Izv. RAN Ser. Matem., 66:5 (2002), 33–82 | MR | Zbl

[34] D. P. Il'utko, “Branching extremals of the length functional in a $\lambda$-normed space”, Matem. sbornik, 197:5 (2006), 75–98 | DOI | MR

[35] K. J. Swanepoel, “The local Steiner problem in normed planes”, Networks, 36:2 (2000), 104–113 | 3.0.CO;2-K class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[36] Sb. Math., 197:9 (2006), 1309–1340 | DOI | DOI | MR | Zbl

[37] Moscow University Math. Bull., 64:2 (2009), 62–66 | DOI | MR

[38] A. O. Ivanov, A. A. Tuzhilin, One-dimensional Gromov minimal filling, 2011, arXiv: 1101.0106v2[math.MG]

[39] Sbornik: Mathematics, 203:5 (2012), 677–726 | DOI | DOI | MR | Zbl