Tverberg Partitions of Points on the Moment Curve

Define M d = { z ( t ) : t ∈ R } , where z ( t ) = ( t , t 2 , … , t d ) ∈ R d . Suppose A = { z ( t i ) : 1 ≤ i ≤ n } ⊂ M d , where t 1 < t 2 < ⋯ < t n . We show that the set A is “usually” in “strong general position” (SGP). The alternating r -partition of A is ( A 1 , A 2 , … , A r ) , where We observe that if r = 2 and n ≥ d + 2 , then conv A 1 ∩ conv A 2 ≠ ∅ (i.e., ( A 1 , A 2 ) is a Radon partition of A ). For r ≥ 3 we show that if n ≥ T ( d , r ) ( = ( d + 1 ) ( r - 1 ) + 1 ) , then ⋂ ν = 1 r conv A ν ≠ ∅ , provided the numbers t 1 , t 2 , … , t n are chosen “sufficiently far”. As a consequence, if n ≥ L ( d , r , k ) = T ( d , k ) + ( r - k ) ⌈ T ( d , k ) k ⌉ , ( r ≥ 2 , 2 ≤ k ≤ min ( d , r - 1 ) ) , and the numbers t 1 , t 2 , … , t n are chosen sufficiently far, then the alternating r -partition of A is an ( r , k )-partition, i.e., each k of the sets conv A ν ( 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 R d 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.