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.
