For potential functions $f:[-1,1]\rightarrow \mathbb{R}$, satisfying that $f^{(k)}(t)\geq 0$, for $t\in(-1,1)$ and all $0\leq k \leq m$, and $f^{(m+1)}(t)<0$ for $t\in(-1,1)$, is it true that a minimizing probability measure for the energy

$$I(\mu)=\int_{\mathbb{S}^2}\int_{\mathbb{S}^2}f(\langle x,y\rangle) d\mu(x)d\mu(y)$$

must be finitely supported? This is true for $m=0$, for in this case the energy is uniquely minimized by a delta mass (since the potential function is decreasing). One can also show it holds for $m=2,3,$ and $5$ via a (non-trivial) linear programming argument, and that the regular simplex, octahedron, and icosahedron uniquely (up to orthogonal transforms) serve as minimizers as each of these configurations are tight spherical designs.

The analogous statement is true on $\mathbb{S}^1$ where appropriately equally spaced points will always appear uniquely as the support of minimizing measures.