Cyclic Polytopes and Oriented Matroids

Abstract

Consider the moment curve in the real euclidean space R d defined parametrically by the mapis the convex hull of n > d different points on this curve. The matroidal analogs are the alternating oriented uniform matroids. A polytope (resp. matroid polytope) is called cyclic if its face lattice is isomorphic to that of C d (t 1 , . . . , t n ). We give combinatorial and geometrical characterizations of cyclic (matroid) polytopes. A simple evenness criterion determining the facets of C d (t 1 , . . . , t n ) was given by Gale [21]. We characterize the admissible orderings of the vertices of the cyclic polytope, i.e., those linear orderings of the vertices for which Gale's evenness criterion holds. Proofs give a systematic account on an oriented matroid approach to cyclic polytopes.

On multiple-instance learning of halfspaces

“…The following lemma is referred to as "unpublished 'folklore'" and proven in an oriented matroid version by Cordovil and Duchet [4]. 1 It is also given as an exercise in Matoušek [11], and it is proven here for completeness.…”

“…1 It is also given as an exercise in Matoušek [11], and it is proven here for completeness. [4,11].) Every set A ⊆ R d of R(d + 1, n) points in general position contains n points such that their convex hull is combinatorially equivalent to d-dimensional cyclic polytopes on n vertices.…”

A Sufficient Condition for the Existence of Large Empty Convex Polygons

“…. , k so that a triple {q 1 , q 2 , q 3 } ∈ Q 3 is colored by color i if the interior of the triangle q 1 q 2 q 3 contains exactly i points of P. By Ramsey's theorem, Q contains a monochromatic subset R of size (log log|Q|) = (log log N ) = (log [2] N ) (log [t] denotes t iterations of log). Every triangle determined by R contains a constant number, c, of points of P in the interior.…”

Problems and Results around the Erdös-Szekeres Convex Polygon Theorem

“…First we recall the following strengthening of the Erdős-Szekeres theorem, which seems to be folklore. See [13] or [11,Proposition 9.4.7] for a proof. Note that in the above theorem we cannot replace the cyclic polytopes with any class of polytopes of different combinatorial kind: one may select any number of points on the moment curve yet every n-element subset will determine a cyclic polytope.…”