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

Bárány, Imre and Matousek, J. and Pór, Attila (2016) Curves in R-d intersecting every hyperplane at most d+1 times. JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY, 18 (11). pp. 2469-2482. ISSN 1435-9855

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Abstract

By a curve in R-d we mean a continuous map gamma : I -> R-d, where I subset of R is a closed interval. We call a curve gamma in R-d (<= k)-crossing if it intersects every hyperplane at most k times (counted with multiplicity). The (<= 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 is an element of[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 R-d 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 R-d concerning order-type homogeneous sequences of points, investigated in several previous papers.

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