Definition convex sets

Question

$C$ is convex $\Longrightarrow \forall x,y \in C$ and $\forall$ $t\in [0,1], \space(1 − t ) x + t y \in C$

My question how comes this formula $(1 − t ) x + t y$ describes all the elements in $[x,y]$ with $t\in [0,1]$?

Answer 1

It doesn't describe all the elements. All this definition says, is that all the points on the line segment from $x$ to $y$ also lie in $C$.

EDIT (after the question was changed): You have two points $x,y$ in $C$, and the equation for the linear space with direction $y-x$ is $t(y-x)$, for $t\in\mathbb R$. But what you want is the affine line parallel to this linear space and which passes through $x$: the equation is $x+t(y-x)$, for $t\in\mathbb R$. Finally, you don't want the whole affine space, just the segment between $x$ and $y$: $x+t(y-x)=(1-t)x+ty$, for $t\in[0,1]$.