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.