Convex Geometry - Key Properties

Convex sets possess special properties enabling powerful theorems with diverse applications.

DefinitionCaratheodory's Theorem

Every point in $\text{conv}(S) \subseteq \mathbb{R}^n$ can be written as a convex combination of at most $n+1$ points from $S$ .

This bounds the complexity of convex hulls: in the plane, every point is a combination of at most 3 points; in 3D, at most 4 points suffice.

Radon's theorem states that any set of $n+2$ points in $\mathbb{R}^n$ can be partitioned into two sets whose convex hulls intersect. This combinatorial result has topological applications (Borsuk-Ulam theorem).

DefinitionHelly's Theorem

If $\{C_1, \ldots, C_m\}$ is a finite family of convex sets in $\mathbb{R}^n$ with $m \geq n+1$ , and every $(n+1)$ of them have non-empty intersection, then all $m$ sets have non-empty intersection:

\bigcap_{i=1}^{n+1} C_i \neq \emptyset \text{ for all choices} \implies \bigcap_{i=1}^m C_i \neq \emptyset

Helly's theorem has numerous applications in computational geometry, optimization, and statistics. It characterizes when families of convex sets intersect based on local conditions.

ExampleWidth and Diameter

The width of a convex body $C$ in direction $u$ is:

w_u(C) = \max_{x \in C} \langle x, u \rangle - \min_{x \in C} \langle x, u \rangle

The minimum width over all directions is the thickness of $C$ . Constant width bodies (like Reuleaux triangles) have applications in mechanical engineering.

Remark

The Brunn-Minkowski inequality relates volumes of convex sets and their Minkowski sums:

V(C + D)^{1/n} \geq V(C)^{1/n} + V(D)^{1/n}

This fundamental inequality implies isoperimetric inequalities and has applications in geometric measure theory, probability, and information theory.

Affine transformations preserve convexity, making many results coordinate-independent. The study of affine-invariant properties forms affine convex geometry, complementing metric-based approaches.