According to the definition


(Support function)

Note that
is a positively homogenous function of
.
Suppose
is some proper convex positively homogenous function. Consider the
conjugate
By the positive homogeneity
.
Consequently, for any
,
Hence,
is either 0 or
.
Introduce the
set
Such set is a
.
Indeed, if
then
and 0 is reached by scaling with
.
Then
.
On the other hand if
then
and
is reached by scaling.
The last part of the summary follows from the result
(
Conjugate duality theorem
). We
established onetoone correspondence between convex sets and proper convex
positively homogenous functions.
