Minimizing $|XA|^3 + |XB|^3 + |XC|^3$ for a triangle $\triangle ABC.$

Question

I was just curious - is there a closed form for: $$\min_{X} |XA|^3 + |XB|^3 + |XC|^3$$ for a given triangle $\triangle ABC$ ? This problem is well-known for $n = 1:$ $$\min_{X} |XA| + |XB| + |XC| = \sqrt{a^2+b^2+c^2+2\sqrt{3}S}$$ where $S$ is the area of $\triangle ABC$ and it is attained when $X$ is the Fermat-Toricelli point (let's just focus on acute $\triangle ABC$ for simplicity). For $n = 2:$ $$\min_{X} |XA|^2 + |XB|^2 + |XC|^2 = \dfrac{a^2+b^2+c^2}{3}$$ and attained when $X$ is the centroid $G$ and its proof is a cute vector trick by writing $|XA|^2 = (\vec{XG} + \vec{GA}, \vec{XG} + \vec{GA}).$ It's clear that such an optimal $X$ must be on the same plane as $\triangle ABC$ by simply projecting.