Convex set vs. convex function
“Convex” describes both sets and functions, and the two meanings are linked. A convex set contains the segment between every two of its points, and a convex function lies on or below the segment between every two points on its graph. A function is convex exactly when the region on and above its graph is a convex set.
Definitions
Level:
Missing definition: convex_set
Missing definition: convex_function
Examples
| Example | convex_set | convex_function | Why |
|---|---|---|---|
| The closed unit disk $D = \setdef{(x, y) \suchthat x^2 + y^2 \leq 1}$ | ✓ | – | |
| The ring $\setdef{(x, y) \suchthat 1 \leq x^2 + y^2 \leq 4}$ (an annulus) | ✗ | – | |
| $f(x) = x^2$ on $\reals$ | – | ✓ | The squaring function, as a function $f : \reals \to \reals$. |
| $f(x) = \abs{x}$ | – | ✓ | |
| $x \mapsto \sin x$ | – | ✗ |