Lines, betweenness and metric spaces

Abstract:

A celebrated Theorem of de Bruijn and Erdos states that every noncollinear set of n points in the plane determines at least n distinct lines. Line uv in the plane consists of all points p such that:

– dist(p,u) + dist(u,v) = dist(p,v) (i.e. u is between p and v) or

– dist(u,p) + dist(p,v) = dist(u,v) (i.e. p is between u and v) or

– dist(u,v) + dist(v,p) = dist(u p) (i.e. v is between u and p).

With this definition of line uv in an arbitrary metric space (V, dist), Chen and Chvatal conjectured that every metric space on n points, where n is at least 2, has at least n distinct lines or a line that consists of all n points. The talk will survey results supporting this conjecture as well as some discussions around lines induced by a set of points together with a betweenness relations. Most of the presented results can be found in “Lines, betweenness and metric paces” (P.A., X. Chen, G.

Huzhang, R. Kapadia, C. Supko).

Posted on Mar 31, 2015 in Discrete Mathematics, Seminars