Solved it.

Let $S$ be the Least Core of $(\Pi,F)$. Let $x_1,x_2 \in S$. Show for an arbitrary $\alpha \in (0,1) \,\alpha x_1 + (1-\alpha) x_2$ is contained in $S$.

$x_1,x_2\in S$ so $\bigvee_{j\in M}F_j(x_1)\leq\bigvee_{j\in M}F_j (y)\,, \forall y\in \Pi$ and $\bigvee_{j\in M}F_j(x_2)\leq\bigvee_{j\in M}F_j (y)\,, \forall y\in \Pi$

Because the $F_j$'s are convex we know$F_j(\alpha x_1+(1-\alpha )x_2)\leq \alpha F_j(x_1) + (1-\alpha)F_j(x_2)$. Without loss of generality we may assume $\bigvee F_j(x_2)\geq \bigvee F_j(x_1)$, so: $\bigvee F_j(\alpha x_1+(1-\alpha )x_2)\leq \bigvee F_j(x_2)$ and because $x_2$ lies in the least core, so does $\alpha x_1+(1-\alpha )x_2$\\