Cone-separation and star-shaped separability with applications

Let XX be a (real) Banach space, AA be a subset of XX and x∉Ax∉A. We present cone-separation in terms of separation by a collection of linear functionals defined on XX and obtain necessary and sufficient conditions for cone-separability AA and xx. Also, we give characterizations for star-shaped separability. Finally, as an application of separability, we characterize best approximation problem by elements of star-shaped sets.