This theorem can be considered a converse of a previous theorem.
Theorem: Let denote a positive homogenous, subadditive function defined on a linear space over the reals.
(i) The set of points satisfying is a convex subset of , and 0 is an interior point of it.
(ii) The set of points satisfying is a convex subset of .
Proof: (i) Let . Let . For ,
Therefore is convex. We also have .
The proof of (ii) is similar.