Convex function

Definition

Let $I$ be an interval and let $f \colon I \to \mathbb{R}$ be a function. The function $f$ is convex if $$f(t x + (1 - t) y) \le t f(x) + (1 - t) f(y)$$ for all $x, y \in I$ and all $t \in [0, 1]$.

Remarks

Geometrically, each chord of the graph lies on or above the graph. For example, $x^2$, $|x|$, and $e^x$ are convex on $\mathbb{R}$. Linear functions are convex. A twice differentiable function with $f'' \ge 0$ is convex. Mathlib defines convexity on a set as ConvexOn.

Explanations

L1 human

A convex function bends upward, like a bowl. The straight segment between two points of the graph never goes below the graph.