Pseudo strict convexity and metric convexity in metric linear spaces (Q5903670)

Let (X,d) be a real metric linear space. Then d defines a strictly convex norm on X if and only if the following two conditions are satisfied: (i) for distinct x,y in X there exists z (different from x,y) such that \(d(x,z)+d(z,y)=d(x,y)\); and (ii) if x,y are non-zero such that \(d(x+y,0)=d(x,0)+d(y,0)\) then \(y=tx\) for some \(t>0.\) The case of complex linear metric spaces is also considered.