Abstract
Define Md= { z(t) : t∈ R} , where z(t) = (t, t2, … , td) ∈ Rd. Suppose A= { z(ti) : 1 ≤ i≤ n} ⊂ Md, where t1< t2< ⋯ < tn.(a)We show that the set A is “usually” in “strong general position” (SGP).(b)The alternating r-partition of A is (A1, A2, … , Ar) , where [Equation not available: see fulltext.] We observe that if r= 2 and n≥ d+ 2 , then conv A1∩ conv A2≠ ∅ (i.e., (A1, A2) is a Radon partition of A). For r≥ 3 we show that if n≥ T(d, r) (= (d+ 1) (r- 1) + 1) , then ⋂ν=1rconvAν≠∅, provided the numbers t1, t2, … , tn are chosen “sufficiently far”.(c)As a consequence, if (Formula presented.) and the numbers t1, t2, … , tn are chosen sufficiently far, then the alternating r-partition of A is an (r, k)-partition, i.e., each k of the sets conv Aν (1 ≤ ν≤ r) have a point in common. (L(d, r, k) is the smallest n such that a set of n points in SGP in Rd may admit an (r, k)-partition.) In this paper we investigate some relationships among three notions: strong general position, Tverberg’s theorem and the moment curve.
Original language | English |
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Pages (from-to) | 56-70 |
Number of pages | 15 |
Journal | Discrete and Computational Geometry |
Volume | 57 |
Issue number | 1 |
DOIs | |
State | Published - 1 Jan 2017 |
Bibliographical note
Publisher Copyright:© 2016, Springer Science+Business Media New York.
Keywords
- (r, k)-partitions
- Moment curve
- Strong general position
- Tverberg’s theorem