A condition for sets in \(\mathbb{R}^3\) to be cones (Q1961859)

We say that a boundary point \(y\) of a nonempty set \(S\subset\mathbb{R}^3\) is clearly \(R\)-visible from a point \(x\) via \(S\) provided there is a neighborhood \(N\) of \(y\) such that every half-line whose initial point is \(x\) and which has nonempty intersection with \(S\cap N\) is a subset of \(S\). The author proves that if every two boundary points of \(S\) are clearly \(R\)-visible via \(S\) from a point, then \(S\) is a cone.