A characterization of finite Chebyshev sequences in Rn

AbstractGiven an arbitrary totally ordered set Γ, we distinguish three types of sequences A=(ai)i∈Γ⊂Rn, called LI-, H-, and T-sequences according as det[ai1,ai2,…, ain] is non-zero for some i1<i2<…<in, is non-zero for all i1<i2<…<in, and is of constant non-zero sign for all i1<i2<…<in. In this paper we show that if Γ is of finite cardinality, then the inhomogeneous system of linear inequalities <ai, x> = αi, iϵΓ⧹B (α), sgn<ai, x> = sgnαi, iϵB (α), is solvable for all saturated α ≜ (αi)i∈Γ⊂R1 with S+(α) ⩽ n−1 iff A is a T-sequence. Here saturated means that there is a unique way of replacing the zeros of α by + 1 and – 1, in order to reach S+(α). B(α) is the union of those “intervals” of Γ of maximal cardinality ⩾ 2 (called blocks), on which α has a constant non-zero sign

Diszkrét és konvex geometria = Discrete and convex geometry

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

Competition and nonlinear pricing in telecommunications

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