Convex set

Let $X$ be a vector space (typically over $\mathbb{R}$). A set $\Omega\subset X$ is convex if for all $x,y\in\Omega$ and all $\lambda\in[0,1]$ we have

Context. Equivalently, $\Omega$ is convex iff it contains every line segment joining any two of its points. Convexity is the core geometric notion underlying convex analysis and optimization.

Examples:

• Any affine subspace of $\mathbb{R}^n$ (e.g., a line or plane) is convex.
• Any ball $\{x:\|x-x_0\|\le r\}$ in a normed space is convex.
• A halfspace $\{x\in\mathbb{R}^n:\langle a,x\rangle\le b\}$ is convex.

Non-example.

• The annulus $\{x\in\mathbb{R}^2:1<\|x\|<2\}$ is not convex: a segment between two points may pass through the “hole.”