Here is another class of examples, namely surfaces containing a rational curve with a "very negative" normal bundle.

Let us consider a surface of degree $d$ in $\mathbb{P}^3$, containing a line $L$, and let $f \colon L \to X$ be the inclusion. By the adjunction formula $K_X= \mathcal{O}_X(d-4)$, so $K_XL=d-4$. It follows that 
$L^2=2-d$, that is

$N_{L|X}=\mathcal{O}_{P^1}(2-d)$.

Now let us assume $d \geq 5$, so that $X$ is a surface of general type. By the short exact sequence

$0 \to T_L \to (f^*T_X) \otimes \mathcal{O}_L \to N_{L|X} \to 0$

it follows 

$H^1(L, f^*T_X) \to H^1(L, N_{L|X}) \cong \mathbb{C}^{d-3} \to 0$,

that is $X$ is not convex. On the other hand, since $X$ is of general type, the group $\textrm{Aut}(X)$ is finite, so $X$ is certainly not homogeneous.