Convex function

A convex function is a continuous function whose value at the midpoint of every interval in its domain does not exceed the arithmetic mean of its values at the ends of the interval.

A function $f: I \mapsto \mathbb{R}$ for some interval $I \subseteq \mathbb{R}$ is convex (sometimes written concave up) over $I$ if and only if the set of all points $(x,y)$ such that $y \ge f(x)$ is convex. Equivalently, $f$ is convex if for every $\lambda \in [0,1]$ and every $x,y \in I$,

$\lambda f(x) + (1-\lambda)f(y) \ge f\left( \lambda x + (1-\lambda) y \right)$.

We say that $f$ is strictly convex if equality occurs only when $x=y$ or $\lambda \in \{ 0,1 \}$.

Usually, when we do not specify $I$, we mean $I = \mathbb{R}$.

We say that $f$ is (strictly) concave (or, occasionally, that it is concave down) if $-f$ is (strictly) convex.

If $f$ is differentiable on an interval $I$, then it is convex on $I$ if and only if $f'$ is non-decreasing on $I$. Similarly, if $f$ is twice differentiable over an interval $I$, we say it is convex over $I$ if and only if $f''(x) \ge 0$ for all $x \in I$.

Note that in our previous paragraph, our requirements that $f$ is differentiable and twice differentiable are crucial. For a simple example, consider the function

$f(x) = \lfloor x \rfloor (x - \lfloor x \rfloor ) + {\lfloor x \rfloor \choose 2}$,

defined over the non-negative reals. It is piecewise differentiable, but at infinitely many points (for all natural numbers $x$, to be exact) it is not differentiable. Nevertheless, it is convex. More significantly, consider the function

$f(x) = \left( |x| - 1 \right)^2$

over the interval $[-2, 2]$. It is continuous, and twice differentiable at every point except ${} (0, 1)$. Furthermore, its second derivative is greater than 0, wherever it is defined. But its graph is shaped like a curvy W, and it is not convex over $[-2,2]$, although it is convex over $[-2,0]$ and over $[0,2]$.

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