I was recently explaining to a friend what the convex hull of a set of points is using the analogy of an elastic band around a set of nails hammered into a board. I was about to say that we can generalize this to three-dimensions by replacing the elastic band with shrink wrap, but this is false! For example, if you shrink wrap a dumbell you will get an hourglass shape.
Is there is an appropriate physical process that really does give the convex hull? If the answer is no, is there some reason why we shouldn't expect physically reasonable minimization problems to give convex hulls?