(Directions of recession). Let
be a closed proper convex function.
1. All nonempty level sets
have the same recession cone given
2. If one nonempty level set is compact then all the level sets are compact.
has no common direction of recession then the minimum cannot escape to
infinity. Such intuition may be formalized into a proof by considering
intersections of the nested compact convex sets
with the sequence
converging to the
The following proposition is a consequence of the same observation and the
) and (