[lcc-002Q]

There is an natural way to extend the convex structure of \mathbb {R} to both \lsqb -\infty , \infty \rpar and \lpar -\infty , \infty \rsqb , by the convention that any nontrivial convex combination involving an infinity is equal to that infinity. This also gives the adjectives convex and concave a meaning when applied to functions X \to \lpar -\infty , \infty \rsqb . For example, a function f: X \to \lpar -\infty , \infty \rsqb is convex if and only if the subset where it's finite is a convex subset of X, and it's a convex function in the ordinary sense on this set.

This doesn't work for the extended real line \eRR = \lsqb -\infty , \infty \rsqb , since there is no sensible interpretation of \theta \cdot -\infty + (1-\theta )\infty . We will inescapably meet some functions which take value in the full extended reals, but where we still wish to speak of their convexity (or concavity).

Hence we adopt the convention that a function f: X \to \eRR is convex if it obeys Jensen's inequality whenever it makes sense, i.e whenever we do not have f(x) = -\infty , f(x') = \infty or vice versa.