What kind of maps preserve convexity?

Question

Linear maps seem to preserve convexity of an object. Are there more maps with this property? What subset of projections preserve convexity?

Answer 1

I do not understand what you mean by object, if a set or a function. In the case of a set, note that there is no necessary condition as explained in this answer.

If for object you mean a function, it is easier:

$-f$ is concave if and only if $f$ is convex. Nonnegative weighted sums:

• if $w_{1}, \ldots, w_{n} \geq 0$ and $f_{1}, \ldots, f_{n}$ are all convex, then so is $w_{1} f_{1}+\cdots+w_{n} f_{n}$. In particular, the sum of two convex functions is convex.

• this property extends to infinite sums, integrals and expected values as well (provided that they exist). Elementwise maximum: let $\left\{f_{i}\right\}_{i \in I}$ be a collection of convex functions. Then $g(x)=\sup _{i \in I} f_{i}(x)$ is convex. The domain of $g(x)$ is the collection of points where the expression is finite. Important special cases:

• If $f_{1}, \ldots, f_{n}$ are convex functions then so is $g(x)=\max \left\{f_{1}(x), \ldots, f_{n}(x)\right\}$

• Danskin's theorem: If $f(x, y)$ is convex in $x$ then $g(x)=\sup _{y \in C} f(x, y)$ is convex in $x$ even if $C$ is not a convex set. Composition:

• If $f$ and $g$ are convex functions and $g$ is non-decreasing over a univariate domain, then $h(x)=g(f(x))$ is convex. As an example, if $f$ is convex, then so is $e^{f(x)} .$ because $e^{x}$ is convex and monotonically increasing.

• If $f$ is concave and $g$ is convex and non-increasing over a univariate domain, then $h(x)=g(f(x))$ is convex.

• Convexity is invariant under affine maps: that is, if $f$ is convex with domain $D_{f} \subseteq \mathbf{R}^{m}$, then so is $g(x)=f(A x+b),$ where $A \in \mathbf{R}^{m \times n}, b \in \mathbf{R}^{m}$ with domain $D_{g} \subseteq \mathbf{R}^{n}$.

• Minimization: If $f(x, y)$ is convex in $(x, y)$ then $g(x)=\inf _{y \in C} f(x, y)$ is convex in $x,$ provided that $C$ is a convex set and that $g(x) \neq-\infty$

• If $f$ is convex, then its perspective $g(x, t)=t f\left(\frac{x}{t}\right)$ with domain $\left\{(x, t) \mid \frac{x}{t} \in \operatorname{Dom}(f), t>0\right\}$ is convex.