The exercise: If A is a convex subset of R^n, show that A is contractible.
What I done: Fixing any x_0 in A, let f(x,t) := (1-t)x+t*x_0. f is continuous and for any x_1, f(x_1, t) is the line from x_1 to x_0 in A (i.e., is entirely inside of A by convexivity...convexness?...convexitude?...convale scence?), so f witnesses A to be contractible.
Is that it?