Operations Preserving Convexity

Operations Preserving Convexity: Let $X$ be a vector space and let $f,f_i:X\to\overline{\mathbb{R}}$ be convex functions for $i=1,\dots,m$. Then:

1. (Nonnegative scaling) For any $\lambda\ge 0$, the function $\lambda f$ is convex.
2. (Finite sums) The function $\sum_{i=1}^m f_i$ is convex.
3. (Finite maxima) The function $\max_{1\le i\le m} f_i$ is convex.

These closure properties are foundational for building new convex functions from old ones and are frequently combined with composition rules and supremum constructions .

Proof sketch (idea): Use the Jensen inequality characterization from equivalent characterizations of convex functions and check it termwise for each operation.