Discrete Geometry

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Details of project

Identifier

37752

Type

Principal investigator

Heppes, Aladár

Title in Hungarian

Diszkrét geometria

Title in English

Discrete Geometry

Panel

Mathematics and Computing Science

Department or equivalent

Department of Geometry (Eötvös Loránd University)

Participants

Bezdek, Károly
Böröczky, Károly
Csikós, Balázs
Daróczy Kiss, Endre
Kertész, Gábor
Lángi, Zsolt
Matolcsi, Máté
Visy, Balázs

Starting date

2002-01-01

Closing date

2005-12-31

Funding (in million HUF)

13.116

FTE (full time equivalent)

0.00

state

closed project

Final report

Results in Hungarian

A kutatások folyamán elért új eredmények zömét publikáltuk, illetve közzétételük folyamatban van. A legfontosabb témakörök az alábbiak: Egyoldali Hadwiger szám [5]. Fejes Tóth szoliditási sejtése [12]. Kör/gömb-elhelyezés - Kétféle kört tartalmazó kitöltések [13]. - Minkowski elhelyezések [3], [21]. - Alternatív hatékonysági mértékek [1, 2, 4]. - Tammes probléma (n=13 [9], illetve n=14,15,16 és 17 [7]). - Voronoi celláinak felszínbecslései [24]. Halmazok átmérője Minkowski terek átlósan maximális halmazai [19], Cilindrikus Borsuk-feldarabolások [34]. Jellemzés felület-vágással [23] Petty szám becslés véges normált terekben [6]. Antipodális poliéderek és él-antipodális poliéderek [11, 26] Relatív hossz problémák [16, 17, 18, 22, 28, 36]. Fedési problémák A „nem-keresztezési feltétel” szükségessége [14], illetve az elliptikus sík fedése [15]. Izoperimetrikus problémák mozaikokon [35], geodetikus görbülettű oldalakkal határolt poligonok állandó görbületű síkon [30]. Helly típusú transzverzális problémák [27, 32, 33]. Kneser-Poulsen problémakör [20, 29]. Egyéb publikált eredmények [8], [10], [25], [31].

Results in English

Most of the new results have been published or being published. The most importan topics are as follows: One-sided Hadwiger number [5]. Fejes Tóth’s solidity conjecture [12]. Packing of discs/balls - Two-size packings [13]. - Minkowskian packings [3], [21]. - Alternative efficiency measures [1, 2, 4]. - Tammes’ problem (for n=13 [9], for n=14,15,16 és 17 [7]). - Surface area of Voronoi cells [24]. Diameter of sets Diametrally maximal sets in Minkowskian spaces [19], Cylindrical Borsuk-decomposition [34]. Characterization of sets of constant width by coundary cuts [23] Petty number Estimates in normed spaces [6]. Antipodális poliéderek és él-antipodális poliéderek [11, 26] Relatíve length problems [16, 17, 18, 22, 28, 36]. Covering problems On the possibility of dropping the „non-crossing” assumption [14], the thinnest covering of the elliptic plane by 4 circles [15]. Izoperimetric problems for tilings [35], for polygons bonded by sides of given geodethic curcvature (on planes of constant curvature.) [30]. Helly type transversal problems [27, 32, 33]. Kneser-Poulsen problems [20, 29]. Other published results [8], [10], [25], [31].

Full text

http://real.mtak.hu/337/

Decision

Yes

List of publications

Böröczky K ; Szabó L: Arrangements of 13 points on a sphere, In: Bezdek A (ed.) Discrete Geometry, New York-Basel: Marcel Dekker, 2003. pp. 111-184, 2003

Böröczky K; Szabó L: Minkowski arrangements of circles in the plane, Suppl.Rend.Circ.Mat.Palermo, Ser II/70: 87-92, 2002

Böröczky K ; Lángi Z: On the relative distances of six points in a plane convex body, Stud. Sci. Math. Hungar. 42(3) 253-264, 2005

Bezdek K; Daróczy-Kiss E: Finding the best face on a Voronoi polyhedron - the strong dodecahedral conjecture revisited, Monatshefte fur Math. (elfogadva), 2005

Bezdek K; Naszódi M,: Rigidity of ball-polyhedra in Euclidean 3-space,, European Journal of Combinatorics (elfogadva), 2005

Böröczky K: The Newton-Gregory problem revisited, In: Bezdek A (ed.) Discrete Geometry, New York-Basel: Marcel Dekker, 2003. pp. 103-110, 2003

Böröczky K; Csikós B: Small convex polytops with long edges and many vertices, Discr. Comput. Geom. 30: 195-204, 2003

Bezdek K; Brass P: On k(plus)-neighbour packings and one-sided Hadwiger configurations, Beitrage zur Alg. und Geom. 44/2, 493-498, 2003

Bezdek K; Naszódi M; VisyB: On the m-th Petty numbers of normed spaces, In: Bezdek A (ed.) Discrete Geometry, New York-Basel: Marcel Dekker, 2003. pp. 291-304, 2003

Böröczky ; Szabó L: Arrangements of 14, 15, 16 and 17 points on a sphere, Studia Sci.Math.Hungar. 4Studia Sci.Math.Hungar. 40: 407-7210: 407-721, 2003

Böröczky ; Szabó L: Minkowski arrangements of balls in n-space, Monatshefte d. Math. 141: 11-19, 2004

Csikós B: Edge-antipodal convex polytopes - a proof of Talata's conjecture, In: Bezdek A (ed.) Discrete Geometry, New York-Basel: Marcel Dekker, 2003. pp. 201-205, 2003

Daroczy-Kiss E: A new bound for the minimal diameter of nine coplanar congruent disks, Annales Univ. Sci. Budapest, Sect. Computat.;(elfogadva), 2005

Florian A ; Heppes A: On the non-solidity of some packings and coverings with circles, In: Bezdek A (ed.) Discrete Geometry, New York-Basel: Marcel Dekker, 2003. pp. 279-290, 2003

Heppes A: Some densest two-size disc packings in the plane, Discr. Comput. Geom. 30: 241-262., 2003

Heppes A: Packing of rounded domains on a sphere of constant curvature, Acta Math. Acad.Sci.Hungar.91/3: 245-252, 2001

Heppes A: On the density of 2-saturated lattice packings of discs, Monatsh. Math. 13: 51-66, 2001

Heppes A: Covering the plane with fat ellipses without non-crossing assumption, Discr. Comput. Geom. 29: 477-481, 2003

Heppes A: Expandability radius versus density of a lattice packing, Periodica Math. Hungar. 45/1-2: 65-71, 2002

Heppes A: Az elliptikus sík legritkább fedése négy egybevágó körrel, Mat. Lapok 8-9. (1998-99/1-2) 4-6, 2003

Heppes A: The width of the transversal strips of T(3) families in the plane, Discr. Comput. Geom. 34, 455-461, 2005

Heppes A: New upper bound on the transversal width of T(3)-families of discs, Discr. Comput. Geom. 34, 463-474, 2005

Heppes A; Kuperberg W: Cylindrical partitions of convex bodies, in Combinatorial and Discr. Geom. (Eds. Goodman, Pach, Welzl, MSRI 52, 399-407, 2005

Heppes A; Morgan F: Planar clusters and perimeter bounds, Philosophical Magazine 83/12, 1333-1345, 2005

Lángi Z: On the relative length of sides of convex polygons, Studia Math. Sci. Hungar. 40, 115-120, 2003

Lángi Z: On seven points in the boundary of a plane convex body in large relative distances, Beitr. Alg. Geom., 45(1): 275-281, 2004

Lángi Z: Relative distance and a convex body touched by its homothetical copies, Bull. Pol. Ac.:Math., 51(4): 439-444, 2003

Lángi Z; Lassak M: On four points of a convex body in large relative distances, Geombinatorics XII(4): 184-190, 2003

Lángi Z; Lassak M: Relative distance and packing a body by homothetical copies, Geombinatorics (elfogadva), 2005

Naszódi M; Visy B: Sets with a unique extension to a set of constant width, Discr. Comput. Geom.: 373-380, 2003

Bezdek; R. Connelly; B. Csikós.: On the perimeter of the intersection of congruent disks., Beiträge zur Algebra und Geometrie, (accepted), 2004

Bezdek, K.; T. Bisztriczky; B. Csikós; A. Heppes: On the transversal Helly numbers of disjoint and overlapping discs, Archiv d. Math. (accepted), 2005

Bezdek, K.; M. Naszódi; D. Oliveros-Braniff,: Antipodality in hyperbolic space., Journal of Geometry (accepted), 2005

Csikós, B.: A Schläfli-type formula for polytopes with curved faces and its application to the Kneser-Poulsen conjecture., Monatsh. für Math., 2005. (accepted)., 2005

Csikós, B.; Zs. Lángi; M. Naszódi.: A generalization of the discrete isoperimetric inequality for piecewise smooth curves of constant geodesic curvature., Periodica Math. Hung. (accepted)., 2005

Averkov, G.,A. Heppes,: Constant Minkowkian width in terms of boundary cuts constant geodesic curvature., Rev. Roumaine Math. Pures 50 (2005), 5-6, 423-429., 2005

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