Curves in Rd intersecting every hyperplane at most d 1 times

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F16%3A10332297" target="_blank" >RIV/00216208:11320/16:10332297 - isvavai.cz</a>

Výsledek na webu

<a href="http://dx.doi.org/10.4171/JEMS/645" target="_blank" >http://dx.doi.org/10.4171/JEMS/645</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.4171/JEMS/645" target="_blank" >10.4171/JEMS/645</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Curves in Rd intersecting every hyperplane at most d 1 times

Popis výsledku v původním jazyce

By a curve in Rd we mean a continuous map γ: I -> Rd, where I is a subset of R is a closed interval. We call a curve γ in Rd (< k+1)-crossing if it intersects every hyperplane at most k times (counted with multiplicity). The (< d+1)-crossing curves in Rd are often called convex curves and they form an important class; a primary example is the moment curve {(t, t2, ..., td): t in [0, 1]}. They are also closely related to Chebyshev systems, which is a notion of considerable importance, e.g., in approximation theory. Our main result is that for every d there is M = M(d) such that every (< d + 2)-crossing curve in Rd can be subdivided into at most M(< d+1)-crossing curve segments. As a consequence, based on the work of Eliáš, Roldán, Safernová, and the second author, we obtain an essentially tight lower bound for a geometric Ramsey-type problem in Rd concerning order-type homogeneous sequences of points, investigated in several previous papers.

Název v anglickém jazyce

Curves in Rd intersecting every hyperplane at most d 1 times

Popis výsledku anglicky

By a curve in Rd we mean a continuous map γ: I -> Rd, where I is a subset of R is a closed interval. We call a curve γ in Rd (< k+1)-crossing if it intersects every hyperplane at most k times (counted with multiplicity). The (< d+1)-crossing curves in Rd are often called convex curves and they form an important class; a primary example is the moment curve {(t, t2, ..., td): t in [0, 1]}. They are also closely related to Chebyshev systems, which is a notion of considerable importance, e.g., in approximation theory. Our main result is that for every d there is M = M(d) such that every (< d + 2)-crossing curve in Rd can be subdivided into at most M(< d+1)-crossing curve segments. As a consequence, based on the work of Eliáš, Roldán, Safernová, and the second author, we obtain an essentially tight lower bound for a geometric Ramsey-type problem in Rd concerning order-type homogeneous sequences of points, investigated in several previous papers.

Druh

J_x - Nezařazeno - Článek v odborném periodiku (Jimp, Jsc a Jost)

CEP obor

BA - Obecná matematika

OECD FORD obor

—

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2016

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Journal of the European Mathematical Society

ISSN

1435-9855

e-ISSN

—

Svazek periodika

18

Číslo periodika v rámci svazku

11

Stát vydavatele periodika

DE - Spolková republika Německo

Počet stran výsledku

14

Strana od-do

2469-2482

Kód UT WoS článku

000386876900002

EID výsledku v databázi Scopus

2-s2.0-84991687119