I know you can use this fact to prove the Homotopy Theorem but I now want to use the Homotopy Theorem (i.e. If $f\simeq g:X\rightarrow Y$ then $f_∗ =g_∗ : H_∗(X) \rightarrow H_∗(Y )$) to prove this fact.
I understand that $H_*(X)=Z_*/B_*$ and that the set, $X$ is convex if $\forall x, y\in X$ and $\forall t\in[0,1]$, $tx+(1-t)y\in X$. What I don't understand is how to use the Homotopy Theorem as instructed, what $Z_*$ and $B_*$ might look like for any convex set, and how to involve the definition of convex in this problem. Granted, some of these thoughts may not be necessary.
Hint: Prove that a convex set is contractible. More specifically, given a convex set $X \subseteq \mathbb{R}^n$, fix some $x_0 \in X$ and prove that $X \simeq \{ x_0 \}$.