I need to prove $A$ is path-connected and its fundamental group is trivial. Path-connectedness is obvious. I know we already have some posts with proofs for this statement. But when we say we can construct a straight line homotopy between the constant map and any other loops, do we use the fact that $A$ is star convex or $A$ is star convex subset of $\Bbb R^n$?
My proof for trivial fundamental group is: let $[f], [g]$ be two homotopy classes of loops based at $x_0\in A$. Then pick $f_0\in [f]$, $g_0\in [g]$ and construct the straight-line homotopy $H(t,x) = (1-t)f_0(x)+tg_0(x)$. Hence, $f_0$ is homotopic to $g_0$.
In my proof of trivial fundamental group, I don't need any property of $A$? Do we use some property of $A$ to show that $H$ is a homotopy between $f_0$ and $g_0$?