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 $m$-designs).
The analogous statement is true on $\mathbb{S}^1$ where an appropriate number of equally spaced points always appear uniquely as the support of minimizing measures (with equal weights). It is also true for $m=2$ and $3$ for spheres in all dimensions.
Outside the above cases, there are a few other cases which this is known, namely for some values of $m$ and $\mathbb{S}^6$, $\mathbb{S}^7$, $\mathbb{S}^{22}$, and $\mathbb{S}^{23}$. In these dimensions (and for certain values of $m$) certain maximal sized equiangular line configurations along with the $E_8$ and Leech lattice minimal vectors must uniquely be minimizers of the above energies, and so minimizers are discrete.
