My textbook says the following.
We call a set $E\subset R^k$ convex if $$\lambda x+(1-\lambda ) y\in E$$ whenever $x\in E, y\in E, 0<\lambda <1.$
I'm totally at a loss as to what this definition is trying to tell me.
Wikipedia elaborates by saying
In other words, every point on the line segment connecting $x$ and $y$ is in [$E$].
But I still don't know what that has to do with the above definition. Where does this come from?
If it helps think about it like this - in concrete terms
If $\lambda =\frac 1 2$ the point $\frac 12 x +\frac 12 y$ is half way between $x$ and $y$.
If $\lambda =\frac 1 3$ the point $\frac 13 x +\frac 23 y$ is two thirds third of the way from $x$ and $y$.
So as the values of $\lambda$ vary from $1$ to $0$ you get further from $x$ and closer to $y$ along the straight line which joined the two points.
To find a point $0.4$ of the way from $x$ to $y$ start at $x$ and go $0.4$ of the way to obtain $x+0.4(y-x)=0.6x+0.4y=0.6x+(1-0.6)y$