Characterizing starshaped sets by maximal visibility

Let S be a nonempty set in a real topological linear space L, p is an element of S is a point of maximal visibility of S if and only if it admits a neighbourhood N in L such that S-q subset of or equal to S-p for every point q is an element of S boolean AND N, where S-x = {s is an element of S: x is visible from s via S}. For S being either open and connected or the closure of its connected interior, it is shown that the kernel of S is the set of all maximal visibility points of S. Planar examples reveal that the topological assumptions on S are necessary. This substantially strengthens a recent result of Toranzos and Forte Cunto.