Partial Convexity

We study $OC$-convexity, which is defined by the intersection of conic semispaces of partial convexity. We investigate an optimization problem for $OC$-convex sets and prove a Krein–Milman type theorem for $OC$-convexity. The relationship between $OC$-convex and functionally convex sets is studied. Topological and numerical aspects, as well as separability properties are described. An upper estimate for the Carathéodory number for $OC$-convexity is found. On the other hand, it happens that the Helly and the Radon number for $OC$-convexity are infinite. We prove that the $OC$-convex hull of any finite set of points is the union of finitely many polyhedra.