The minimum perimeter convex hull of a subset of a point set

Question

Given n points on the plane. No 3 are collinear.

Given the number k.

Find the subset of k points, such that the convex hull of the k points has minimum perimeter

Answer 1

One possible optimization: You can ignore any subsets whose convex hull contains points that are not in the subset.

Proof:

If your convex hull contains points that are not in your subset, then remove a point from your subset that is on the hull, and replace it with a point in the interior of the hull. This will yield a hull of equal or smaller perimeter.