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.

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