Bárány, Imre and Matousek, J. and Pór, Attila (2016) Curves in Rd intersecting every hyperplane at most d+1 times. JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY, 18 (11). pp. 24692482. ISSN 14359855

Text
1309.1147.pdf Download (429kB)  Preview 
Abstract
By a curve in Rd we mean a continuous map gamma : I > Rd, where I subset of R is a closed interval. We call a curve gamma 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, 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 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 Ramseytype problem in Rd concerning ordertype homogeneous sequences of points, investigated in several previous papers.
Item Type:  Article 

Additional Information:  ZURICH, SWITZERLAND 
Uncontrolled Keywords:  Chebyshev system; moment curve; Convex curve; Order type; Ramsey function; Chebyshev system 
Subjects:  Q Science / természettudomány > QA Mathematics / matematika 
SWORD Depositor:  MTMT SWORD 
Depositing User:  MTMT SWORD 
Date Deposited:  03 Jan 2017 10:55 
Last Modified:  03 Jan 2017 10:55 
URI:  http://real.mtak.hu/id/eprint/44195 
Actions (login required)
Edit Item 