Curves in intersecting every hyperplane at most times

  • Imre Bárány

    Hungarian Academy of Sciences, Budapest, Hungary
  • Jiří Matoušek

    Charles University, Praha, Czech Republic
  • Attila Pór

    Western Kentucky University, Bowling Green, USA

Abstract

By a curve in we mean a continuous map , where is a closed interval. We call a curve in -crossing if it intersects every hyperplane at most times (counted with multiplicity). The -crossing curves in are often called convex curves and they form an important class; a primary example is the moment curve . 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 there is such that every -crossing curve in can be subdivided into at most -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 concerning order-type homogeneous sequences of points, investigated in several previous papers.

Cite this article

Imre Bárány, Jiří Matoušek, Attila Pór, Curves in intersecting every hyperplane at most times. J. Eur. Math. Soc. 18 (2016), no. 11, pp. 2469–2482

DOI 10.4171/JEMS/645