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

Imre Bárány ; Jiří Matoušek ; Attila Pór

Journal of the European Mathematical Society, Tome 18 (2016) no. 11, pp. 2469-2482.

Résumé

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 Eliáš, Roldán, Safernová, 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.

Détail

DOI : 10.4171/jems/645

Classification : 05-XX, 52-XX
Keywords: Ramsey function, order type, convex curve, moment curve, Chebyshev system