Real Convex Functions

Why

We speak of functions which always bends up.¹

Definition

Suppose $X \subset \R $ is a convex set. A function $f: X \to \R $ is convex if

\[ f(tx + (1-t)y) \leq tf(x) + (1-t)f(y) \]

for all $y \in [0,1]$ and $x, y \in X$.

In other words, a real-valued function is a function defined on a convex set of real numbers for which the result of the function on a convex combination of any two points in the domain is smaller than the convex combination of the same length of the value of the function on the endpoints.

$f$ is concave if $-f$ is convex.

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