Proper convex function
for at least one x and
A proper concave function is any function g such that is a proper convex function.
For every proper convex function f on Rn there exist some b in Rn and β in R such that
for every x.
The sum of two proper convex functions is not necessarily proper or convex. For instance if the sets and are convex sets in the vector space X, then the indicator functions and are proper convex functions, but is not convex (unless is convex), and is identically equal to if .