Convex Geometry - Examples and Constructions

Convex geometry provides tools for constructing and analyzing geometric and optimization problems.

ExampleVoronoi Diagrams

Given sites $\{p_1, \ldots, p_k\} \subset \mathbb{R}^n$ , the Voronoi cell of $p_i$ is:

V(p_i) = \{x \in \mathbb{R}^n : \|x - p_i\| \leq \|x - p_j\| \text{ for all } j\}

Each Voronoi cell is a convex polyhedron (intersection of halfspaces). Voronoi diagrams partition space into convex regions, with applications in computational geometry, biology, and materials science.

Delaunay triangulation is the geometric dual of Voronoi diagrams. It maximizes the minimum angle among all triangulations, avoiding "skinny" triangles. This makes it valuable in mesh generation for numerical simulations.

DefinitionMinkowski Sum

The Minkowski sum of sets $A$ and $B$ is:

A + B = \{a + b : a \in A, b \in B\}

If $A$ and $B$ are convex, so is $A + B$ . Minkowski sums model configuration spaces in robotics and arise in image processing (morphological operations).

ExampleLinear Programming

Linear programming optimizes a linear objective over a convex polyhedron:

\min \langle c, x \rangle \quad \text{subject to } Ax \leq b

The feasible region is convex. The simplex algorithm and interior-point methods exploit convexity for efficient solution. Applications include resource allocation, network flows, and game theory.

Convex bodies (compact convex sets with non-empty interior) form the primary objects of study. The Blaschke selection theorem states that any bounded sequence of convex bodies has a convergent subsequence (in Hausdorff metric), giving compactness properties essential for existence proofs.

Remark

Applications in machine learning:

Support vector machines separate data using maximum-margin hyperplanes (convex optimization)
Convex neural network architectures guarantee global optima
Compressed sensing exploits convexity of $\ell^1$ minimization for signal recovery

The Gauss map assigns to each boundary point of a convex body the outward normal direction, mapping to the unit sphere. This connection between convex geometry and spherical geometry yields powerful results like the Minkowski existence theorem.