Georgia Institute of Technology College of Sciences

Given a potential function of three vector arguments, $f(x,y,z)$, which is $O(n)$-invariant, $f(Qx,Qy,Qz)=f(x,y,z)$ for all $Q$ orthogonal, we use semidefinite programming bounds to determine optimizing probability measures for interaction energies of the form $\int\int\int f(x,y,z) d\mu(x)d\mu(y)d\mu(z)$ over the sphere. This approach builds on previous use of such bounds in the discrete setting by Bachoc-Vallentin, Cohn-Woo, and Musin, and is successful for kernels which can be shown to have expansions in a particular basis, for instance certain symmetric polynomials in inner products $u=\langle x,y \rangle$, $v=\langle y,z\rangle$, and $t=\langle z, x \rangle$. For other kernels we pose conjectures on the behavior of optimizers, partially inferred through numerical studies.