Let $E\subseteq {\mathbb{R}}^k$ be a convex set, let $\phi :E\to \mathbb{R},$ and let $\ell :E\to \mathbb{R}$ be affine. If $\phi$ is convex, then $\phi +\ell$ and $\phi -\ell$ are convex. If $\phi$ is concave, then $\phi +\ell$ and $\phi -\ell$ are concave. A linear function is affine, so this covers adding or subtracting a linear function.

TODO: prove this.

Let $D\subseteq {\mathbb{R}}^k$ and $E\subseteq {\mathbb{R}}^m$ be convex sets, let $A:D\to E$ be affine, and let $\phi :E\to \mathbb{R}.$ If $\phi$ is convex, then $\phi \circ A$ is convex on $D.$ If $\phi$ is concave, then $\phi \circ A$ is concave on $D.$

TODO: prove this.

Let $\phi$ be twice differentiable on an open interval $(a,b).$ Then $\phi$ is convex on $(a,b)$ if and only if ${\phi }^{″}(x)\ge 0$ for all $x\in (a,b),$ and concave if and only if ${\phi }^{″}(x)\le 0$ for all $x\in (a,b).$