Q:

Understand the definition of convex metric spaces

I am trying to understand the following definition:

We call a set $E\subset \Bbb R^k$ convex if>$$\lambda x+(1-\lambda)y\in E$$
  Whenever $x\in E, y\in E$ and $0\lt \lambda \lt 1$

Clearly this subset of a Euclidean space must be a metric space, so we are looking at a metric space. Then we are looking at some definition which isn't an in/equality? What does this look like geometrically in a 3D case for example and why?(If this is a possible interpretation)

If someone can give an answer that explains the meaning, it will certainly help me in learning to 'read' the Math here. Is there a geometric interpretation?

A:

For $x,y\in\mathbb{R}^n$, the set $\{\lambda x+(1-\lambda)y\mid 0<\lambda<1\}$ is the open line segment from $x$ to $y$.
Thus your definition says that if $x,y\in E$ (convex), then the entire line segment joining $x$ to $y$ is also in $E$.

