Finite metrics in switching classes

Let image be a symmetric function on a finite set D satisfying image for all image. A switch image of g w.r.t. a local valuation image is defined by image for image and image for all x. We show that every symmetric function g has a unique minimal semimetric switch, and, moreover, there is a switch of g that is isometric to a finite Manhattan metric. Also, for each metric on D, we associate an extension metric on the set of all nonempty subsets of D, and we show that this extended metric inherits the switching classes on D.