Convex set

Definition

A subset $S$ of $\mathbb{R}^n$ is convex if for all $x, y \in S$ and each real number $t$ with $0 \leq t \leq 1$, $$(1 - t)\, x + t\, y \in S.$$ That is, $S$ contains the segment from $x$ to $y$.

Remarks

The same definition works in each real vector space or real affine space. Mathlib's Convex is stated for a module over an ordered semiring. A convex set contains each convex combination $\sum_i w_i x_i$ of its points, an affine combination with $w_i \geq 0$. The empty set and the whole space are convex. Each intersection of convex sets is convex, but a union of convex sets need not be convex. Balls, half-spaces, lines and triangles with their inside are convex.

Explanations

L1 human

A set is convex if you can see each of its points from each other point, with the line of sight inside the set. A disc is convex. A ring is not: the segment between two opposite points of the ring passes through the hole.