Convexity properties of space curves

Abstract

A set S is called if, for all points P, Q of S, the line segment PQ is contained in S. A simple closed planar curve and a simple closed surface are not by this definition, but they are called convex if they are boundaries of sets, and similarly a planar arc is called convex if it is a subset of the boundary of a set. This concept of convexity is ordinarily defined only for planar arcs, but we show that it can also be used in 3D. Points on the boundary of a set--in particular, points of a convex curve or surface--have useful visibility and accessibility properties. We establish some of these properties, and also characterize some special classes of convex space arcs and curves.