Clear visibility, starshaped sets, and finitely starlike sets

Let S be an arbitrary nonempty set in Rd. The following results are true for every k, 0⩽k⩽d: the dimension of ker S is at least k if and only if every countable family of boundary points of S is clearly visible from a common k-dimensional neighborhood in S. Similarly, ker S contains a k-dimensional ε-neighborhood if and only if every countable family of boundary points of S is clearly visible from a common k-dimensional ε-neighborhood in S.In the plane, we have the following results concerning finitely starlike sets: for S an arbitrary nonempty set in R2, S is finitely starlike if every three points of cl S are clearly visible from a common point of S. In case S ⊂−R2 and int cl S∼S=∅, then S is finitely starlike if and only if every three points of S are visible from a common point of S. In each case, the number 3 is best possible.