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.
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Imre Bárány, Jiří Matoušek, Attila Pór, Curves in intersecting every hyperplane at most times. J. Eur. Math. Soc. 18 (2016), no. 11, pp. 2469–2482DOI 10.4171/JEMS/645