Euclidian (planar) TSP asks for a tour with the minimum total length. The problem is known to be NP-hard. I am interested in the variant of finding a closed tour with the minimum enclosed area (assuming that the solution has no crossing edges).

What is known about this problem and its computational complexity? Is the decision problem NP-hard?

Decision problem: Given a set of points on the plane, integer A

Question: Is there a tour that has enclosed area less than A?