Finite metrics in switching classes

Let g:D×D→Rg:D×D→R be a symmetric function on a finite set D   satisfying g(x,x)=0g(x,x)=0 for all x∈Dx∈D. A switch gσgσ of g   w.r.t. a local valuation σ:D→Rσ:D→R is defined by gσ(x,y)=σ(x)+g(x,y)+σ(y)gσ(x,y)=σ(x)+g(x,y)+σ(y) for x≠yx≠y and gσ(x,x)=0gσ(x,x)=0 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.