Journal of the European Mathematical Society

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Volume 18, Issue 11, 2016, pp. 2469–2482
DOI: 10.4171/JEMS/645

Published online: 2016-10-12

Curves in $\mathbb R^d$ intersecting every hyperplane at most $d+1$ times

Imre Bárány[1], Jiří Matoušek and Attila Pór

(1) Hungarian Academy of Sciences, Budapest, Hungary

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

Keywords: Ramsey function, order type, convex curve, moment curve, Chebyshev system

Bárány Imre, Matoušek Jiří, Pór Attila: Curves in $\mathbb R^d$ intersecting every hyperplane at most $d+1$ times. J. Eur. Math. Soc. 18 (2016), 2469-2482. doi: 10.4171/JEMS/645