For another class of examples, consider a surface without rational curves and blow-up a finite number of disjoint points. If $E$ is one of the exceptional divisors and $f \colon E \to X$ is the inclusion in the blow-up, since 

$H^0(E, N_{E|X})=H^1(E, N_{E|X})=0$,

we obtain $H^1(E, f^*T_X)=H^1(E, T_E)=0$, so $X$ is convex. On the other hand, $X$ is not homogeneous, since its automorphism group is not transitive.