Convex Region

analysis topology optimisation

Definition

Convex Region

A set $C$ in a real vector space is called convex if, for every pair of points within the set, every point on the straight line segment that joins them is also within the set.

Formally, $C$ is convex if for all $x,y\in C$ and all $\lambda \in [0,1]$:

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

Properties

• Intersection: The intersection of any collection of convex sets is itself a convex set.
• Functions: A function is a convex function if and only if its epigraph (the set of points on or above the graph) is a convex set.
• Convex Hull: The convex hull of a set $S$ is the smallest convex set that contains $S$.
• Separating Hyperplane Theorem: Given two disjoint convex sets, there exists a hyperplane that separates them.