The relationship between ic-cone-convexness and nearly cone-subconvexlikeness

Properties for convex cones are discussed, and are used to obtain the following results: (1) when the ordering cone has nonempty interior, ic-cone-convexness is equivalent to nearly cone-subconvexlikeness; (2) when the ordering cone has empty interior, ic-cone-convexness implies nearly cone-subconvexlikeness, and a counter-example is given to show that the converse implication is not true.