Curves in Rd\mathbb R^d intersecting every hyperplane at most d+1d+1 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 Rd\mathbb R^d we mean a continuous map γIRd\gamma\:I\to\mathbb R^d, where IRI\subset\mathbb R is a closed interval. We call a curve γ\gamma in Rd(k)\mathbb R^d\: (≤ k)-crossing if it intersects every hyperplane at most kk times (counted with multiplicity). The (d)(≤ d)-crossing curves in Rd\mathbb R^d are often called convex curves and they form an important class; a primary example is the moment curve {(t,t2,,td):t[0,1]}\{(t,t^2,\ldots,t^d):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 dd there is M=M(d)M=M(d) such that every (d+1)(≤ d+1)-crossing curve in Rd\mathbb R^d can be subdivided into at most M(d)M\: (≤ d)-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\mathbb R^d concerning order-type homogeneous sequences of points, investigated in several previous papers.

Cite this article

Imre Bárány, Jiří Matoušek, Attila Pór, Curves in Rd\mathbb R^d intersecting every hyperplane at most d+1d+1 times. J. Eur. Math. Soc. 18 (2016), no. 11, pp. 2469–2482

DOI 10.4171/JEMS/645