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

Abstract

By a curve in R^d we mean a continuous map gamma:I -> R^d, where I is a closed interval. We call a curve gamma in R^d at most k crossing if it intersects every hyperplane at most k times (counted with multiplicity). The at most d crossing curves in R^d are often called convex curves and they form an important class; a primary example is the moment curve {(t,t^2,...,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. We prove that for every d there is M=M(d) such that every at most d+1 crossing curve in R^d can be subdivided into at most M convex curves. As a consequence, based on the work of Elias, Roldan, Safernova, and the second author, we obtain an essentially tight lower bound for a geometric Ramsey-type problem in R^d concerning order-type homogeneous sequences of points, investigated in several previous papers.