The complexity of computing minimum separating polygons

Suppose that S and T are two sets of points in the plane. A separating polygon separating S from T is a simple polygonal circuit P for which every point of S is interior or on the boundary of P, and every point of T is exterior or on the boundary of P. Let D(P) denote the perimeter of a polygon P, that is, the sum of the Euclidean lengths of the edges of P. If P is a separating polygon for S and T with minimum value of D(P) then P is called a minimum separating polygon. The problem of computing a minimum separating polygon has been studied in various forms in connection with applications to computer vision and collision avoidance. In this note we show that the version of the problem where S and T are finite sets of points is NP-hard.