Q:

Why does a convex set have trivial fundamental group?

Why does a convex set have trivial fundamental group?
Without using the definition of contractible spaces, could anyone explain for me why that is true?  

A:

Take a loop $\gamma :[0,1]\to C$ based at $x_0\in C$. Define the homotopy
$$H:[0,1]\times [0,1]\to C$$
by the formula
$$H(t,s)=sx_0+(1-s)\gamma(t).$$
This is well defined because $C$ is convex and is is a homotopy between any path $\gamma$ and the constant loop $x_0$, so the fundamental group of $C$ is trivial.

