REAL

Diszkrét és konvex geometria = Discrete and convex geometry

Makai, Endre and Bezdek, András and Böröczky, Károly and Fejes Tóth, Gábor and Katona, Gyula and Pach, János and Pór, Attila and Uhrin, Béla (2008) Diszkrét és konvex geometria = Discrete and convex geometry. Project Report. OTKA.

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Abstract

R^3-ben n pont meghataroz legalabb const n^{77/141 -epsilon} tavolsagot (epsilon >0 tetsz.) R^3-ben n nem koplanaris pont (n>6 paratlan) meghataroz legalabb 2n-5 iranyt, es ez minden fenti n-re pontos. Egy racsteglatestnek, amelybe teljes n-es graf belerajzolhato, hogy a csucsok racspontok, es az elek mas csucson nem mennek at, minimalis terfogata const n^{3/2}. R^3-ben C^2 konvex testeknek korlatozott elszamu konvex poliederekkel torteno terfogatapproximaciojat visgaltuk, es erre aszimptotikus formulat adtunk. Minden harmadfoku graf egyenes elekkel sikba rajzolhato ugy hogy el nem tartalmaz mas csucsot, es az elek iranyai szama legfeljebb const. Az egysegkor veges sok konvex tartomanyra bontasa eseten ezek beirt korei sugarai osszege legalabb 1. R^n-ben egy 2 atlagszelessegu konvex test kore irt szimplex atlagszelessege legalabb akkora mint az egyseggomb kore irt szabalyos szimplexe. R^n-ben (n>1) ket konvex test, amelyek barmely kongruens peldanyainak metszete/uniojuk konvex burka centralszimmetrikus, kongruens gombok. Minden veges sikbeli ponthalmazban van Hamilton-ut, hogy egyik szog sem kisebb 20 foknal. R^n-ben 0-ra csillagszeru testet meghataroznak a linearis (n-1)-alterekkel valo metszetei teruletei es sulypontjai. Fix k-ra n pontu, gorbevonalakkal sikbarajzolt grafokra, amelyeknel nincs k paronkent metszo el, az elszamra a korabbiaknal sokkal jobb felso becslest adtunk. | In R^3 n points determine at least const n^{77/141-epsilon} distances (epsilon >0 arbitrary). In R^3 n not coplanar points (n>6 odd) determine at least 2n-5 directions, sharp for each above n. In R^3 lattice rectangular box, in which complete graph on n vertices can be drawn, vertices being lattice points, edges not containing other vertices, has minimal volume const n^{3/2}. In R^3 we investigated volume approximation of C^2 convex bodies by convex polyhedra with number of edges bounded above, gave asymptotic formula. Each 3rd degree graph can be drawn in R^2 with straight edges, no edge containing other vertices, number of directions of edges bounded. Decomposing the unit disc to finitely many convex domains, sum of the inradii is >= 1. In R^n average width of simplex, circumscribed to convex body of constant width 2, is >= that of regular simplex circumscribed to unit ball. In R^n (n>1) two convex bodies, intersection/convex hull of union of any congruent copies of which being centrally symmetric, are congruent balls. Finite set in R^2 has Hamilton line, each angle >= 20 degrees. In R^n body starlike w.r.t. 0 is determined by areas and barycentres of its sections with linear (n-1)-subspaces. We gave, for fixed k, for edge numbers of graphs with n vertices, drawn in R^2 with curvilinear edges, having no k pairwise intersecting edges, estimates from above, much better than earlier ones.

Item Type: Monograph (Project Report)
Uncontrolled Keywords: Matematika
Subjects: Q Science / természettudomány > QA Mathematics / matematika
Depositing User: Mr. Andras Holl
Date Deposited: 08 May 2009 11:00
Last Modified: 30 Nov 2010 19:40
URI: http://real.mtak.hu/id/eprint/1113

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