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

Abstract

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.