Convex function
acting from
to
.
Level sets
.

The consideration of this entire chapter on convex analysis is restricted to
proper functions. Hence, all functions that are said to be convex are also
presumed to be proper.
Proposition
(Main property of convex
function). A function
is convex iff
.
Proposition
A smooth function
is convex iff the matrix of second derivatives is nonnegatively determined.
