Convex Geometry - Core Definitions

Convex geometry studies convex sets and their properties—fundamental objects appearing throughout mathematics, optimization, economics, and physics.

DefinitionConvex Set

A set $C \subseteq \mathbb{R}^n$ is convex if for any two points $x, y \in C$ and any $\lambda \in [0,1]$ :

\lambda x + (1-\lambda)y \in C

Equivalently, $C$ contains all line segments between its points. Examples include balls, simplices, halfspaces, and polyhedra.

The convex hull $\text{conv}(S)$ of a set $S$ is the smallest convex set containing $S$ . It equals the set of all convex combinations:

\text{conv}(S) = \left\{ \sum_{i=1}^k \lambda_i x_i : x_i \in S, \lambda_i \geq 0, \sum \lambda_i = 1 \right\}

DefinitionExtreme Points

A point $x \in C$ is an extreme point if it cannot be written as a non-trivial convex combination of other points in $C$ . Equivalently, $x$ is not an interior point of any line segment in $C$ .

The set of extreme points is denoted $\text{ext}(C)$ . For a polytope (convex hull of finitely many points), extreme points are the vertices.

ExamplePolytopes

A polytope is the convex hull of finitely many points. In $\mathbb{R}^2$ , polytopes are polygons. In $\mathbb{R}^3$ , they're polyhedra.

Standard examples:

Simplex: Convex hull of $n+1$ affinely independent points in $\mathbb{R}^n$
Cube: $[-1,1]^n$
Cross-polytope: Convex hull of $\pm e_i$ (standard basis vectors)

DefinitionSupporting Hyperplane

A hyperplane $H$ supports convex set $C$ at point $x \in C$ if $C$ lies entirely in one closed halfspace determined by $H$ , and $x \in H$ .

Every boundary point of a closed convex set has a supporting hyperplane (separation theorem consequence).

Remark

Convex geometry underlies optimization theory. Linear programming minimizes linear functions over convex polyhedra. Convex optimization exploits convexity to guarantee global optima and efficient algorithms. Applications span engineering, finance, machine learning, and operations research.

Convex sets exhibit remarkable regularity. Projections of convex sets are convex, intersections preserve convexity, and Minkowski sums (pointwise addition) of convex sets are convex. This stability under operations makes convex geometry tractable.