Finite metrics in switching classes

Let D be a finite set and g : D × D → R a symmetric function satisfying g(x, x) = 0 and g(x, y) = g(y, x) for all x, y ∈ D. A switch gσ is obtained from g by using a local valuation σ : D → R: gσ(x, y) = σ(x) + g(x, y) + σ(y) for x 6= y. It is shown that every symmetric function g has a unique minimal pseudometric switch, and, moreover, there is a switch gσ of g that is isometric to a finite Manhattan metric. Also, for each metric on a finite set 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.