A New Characterization of L-Convex Polygons

In 1949 Horn and Valentine [HV] showed that if each pair of points a,b in a simple polygon P could be connected by a polygonal path of length two lying in P (such polygons are termed L-convex polygons) then through each point x in P there exists a line segment L(x) lying in P such that for every point y in P there exists a point z in L(x) with the property that the segment yz lies in P. Since the converse also holds true this is in fact a characterization of L-convex polygons. We show that by relaxing L(x) from a line-segment to a star-shaped subset S(x) of P containing x we obtain a new characterization of L-convex polygons if S(x) is constrained to be star-shaped from x, and a new class of polygons if it is not.