roposition
For any convex function
is the intersection of the upper half planes.

Corollary
The set
is equal to intersection of the upper halfplanes defined by the hyperplanes
from proof of the previous statement.
Let us denote
.
Let us introduce a
set
Such set is not empty and it is an epigraph of some function because if
then
for all
.
Let us denote such function
.
By
definition
Geometrical meaning of
.

Let us introduce
Observe that
.
Indeed,
means that
or
where the
and
run through all the hyperplanes that define
.
For the same reason the infimum of such
is the
:
The closure part:
comes from the fact that taking affine envelopes includes boundary points of
the
into the final result
.
