Local Steiner convexity

Local Polynomial Convexity

We begin with the following question: given a closed disc D ⋐ C and a complex-valued function F ∈ C(D), is the uniform algebra on D generated by z and F equal to C(D) ? This question is complicated by the presence of points in the surface S := graph D (F) that have complex tangents. Such points are called CR singularities. Let p ∈ S be a CR singularity at which the order of contact of the tange...

On local convexity in graphs

A set K of nodes of a graph G is geodesically convex (respectively, monophonically convex) if K contains every node on every shortest (respectively, chordless) path joining nodes in K. We investigate the classes of graphs which are characterized by certain local convexity conditions with respect to geodesic convexity, in particular, those graphs in which balls around nodes are convex, and those...

Local convexity on smooth manifolds

In the paper, some properties of the spaces of paths are studied in order to define and characterize the local convexity of sets belonging to smooth manifolds and the local convexity of functions defined on the local convex sets of smooth manifolds.

Local improvement in Steiner trees

LOCAL CONVEXITY AND Ln SETS1

We will prove that a closed connected set S in ET whose points of local nonconvexity can be decomposed into a finite number of convex parts also possesses a simple type of polygonal connectedness. Furthermore, if S has a finite number of points of local nonconvexity, we obtain another result. In order to describe this situation easily we use the notation of Ln set used by Horn and Valentine [l ...