Curves in Rd intersecting every hyperplane at most d + 1 times

Abstract

By a curve in Rd we mean a continuous map γ: I → Rd, where I ⊂ R is a closed interval. We call a curve γ in Rd (≤ k)-crossing if it intersects every hyperplane at most k times (counted with multiplicity). The (≤ d)-crossing curves in Rd are often called convex curves and they form an important class; a primary example is the moment curve {(t, t2, …, td): t ∈ [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 Rd can be subdivided into at most M(≤ d)-crossing curve segments. 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 Rd concerning order-type homogeneous sequences of points, investigated in several previous papers.