Convex sets are simply connected

Let $C$ be a convex set. First, note that $C$ is path-connected, since any two points are connected by the segment between them.
Now, let $\gamma_{0},\gamma_{1}$ be two loops based at point $x_{0}$ of $C$.
We can naturally form the following homotopy (by the way, this is always the first homotopy to try!):
$$H(t,s)=(1-t)\gamma_{0}(s)+t\gamma_{1}(s)$$This is indeed well-defined, with values in $C$, since $C$ is convex.
Furthermore, $H$ is a path homotopy, because it preserves the endpoints:
$$H(t,0)=H(t,1)=x_{0}.$$

We have just shown that there is only one homotopy class of loops in $C$. In other words, the fundamental group of $C$ contains only one element, so it is the trivial group.