1 research outputs found
Codimension two and three Kneser Transversals
Let be integers with and let
be a finite set of points in . A -plane
transversal to the convex hulls of all -sets of is called Kneser
transversal. If in addition contains points of , then
is called complete Kneser transversal.In this paper, we present various
results on the existence of (complete) Kneser transversals for .
In order to do this, we introduce the notions of stability and instability for
(complete) Kneser transversals. We first give a stability result for
collections of points in with
and . We then present a description of
Kneser transversals of collections of points in
with for . We show that
either is a complete Kneser transversal or it contains
points and the remaining points of are matched in pairs in
such a way that intersects the corresponding closed segments determined by
them. The latter leads to new upper and lower bounds (in the case when and ) for defined as the maximum positive integer
such that every set of points (not necessarily in general position) in
admit a Kneser transversal.Finally, by using oriented matroid
machinery, we present some computational results (closely related to the
stability and unstability notions). We determine the existence of (complete)
Kneser transversals for each of the different order types of
configurations of points in