📂Linear Algebra

Convex Sets in Vector Spaces

Definition

A subset $M$ of a vector space $V$ is called a convex set if the following equation holds:

$$\lambda x +(1-\lambda)y \in M,\quad \forall \lambda\in[0,1],\ \forall x,y \in M$$

Description

Verbally, this equation means "$M$ is a convex set implies that every vector lying between any two vectors in $M$ also belongs to $M$". Also, if $M$ is a subspace, it is closed under addition and scalar multiplication, making it a convex set.