A Characterization Theorem and An Algorithm for A Convex Hull Problem

We refer to the point p′ satisfying (1) as an -approximate solution. Clearly, approximation to a prescribed absolute error is also possible. We refer to a point p′ satisfying (2) as witness. This condition holds if and only if p 6∈ conv(S). This is because in this case we can prove the Voronoi cell of p′ with respect to the two point set {p, p′} contains conv(S) (see Figure 1). Equivalently, the orthogonal bisector of the line segment pp′ separates p from conv(S).