Curves in Rd intersecting every hyperplane at most d 1 times

Result code in 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>
Result on the web
<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>

Result language
angličtina
Original language name
Curves in Rd intersecting every hyperplane at most d 1 times
Original language description
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.
Czech name
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Czech description
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Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
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Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2016
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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