Convex Sets

Why

We generalize convex sets to arbitrary vector spaces.

Definition

Suppose $V$ is a vector space over the real field $\R $. A set $C \subset V$ is convex if it contains the closed line segment between every pair of distinct points.

In other words, a set $C \subset V$ is convex if

\[ \lambda x + (1-\lambda )y \in C \quad \text{for all } x, y \in C \text{ and } \lambda \in [0,1]. \]