he following statement is direct consequence of definitions.
Proposition
(Infimal convex function)Let
be a convex set. The
function
is convex.
Definition
(Infimal convolution) The function
is called infimal convolution.
Proposition
If
are convex functions then
is a convex function.
Proposition
(Duality of infimal convolution and addition). Let
be convex functions.
Then
