Separation Problems in the Context of h-Convexity

The concept of h-convexity extends the notion of classical convexity, using some nonnegative function h of the weights on the right-hand side in the defining inequality. The aim of the present note is to show that nonconstant h-affine functions appear only in the classical convexity case. Affine and convex separation problems are also studied, without any further restrictions on h. The obtained results suggest that, in view of Convex Geometry or Convex Analysis, the notion of h-convexity may have only particular importance.