Supremum of Convex Functions

Supremum of Convex Functions: Let $X$ be a vector space and let $\{f_i:X\to\overline{\mathbb{R}}\}_{i\in I}$ be a family of convex functions indexed by a nonempty set $I$. Define

Then $f$ is convex on $X$.

This extends the “finite maximum” case in operations preserving convexity and is used heavily to construct convex envelopes and support functions.

Proof sketch (idea): For each $i$, Jensen gives $f_i(\lambda x+(1-\lambda)y)\le \lambda f_i(x)+(1-\lambda)f_i(y)\le \lambda f(x)+(1-\lambda)f(y)$. Taking $\sup_i$ on the left yields Jensen for $f$.