  This paper is devoted to spherical measures and point configurations
optimizing three-point energies. Our main goal is to extend the classic
optimization problems based on pairs of distances between points to the context
of three-point potentials. In particular, we study three-point analogues of the
sphere packing problem and the optimization problem for $p$-frame energies
based on three points. It turns out that both problems are inherently connected
to the problem of nearly orthogonal sets by Erd\H{o}s. As the outcome, we
provide a new solution of the Erd\H{o}s problem from the three-point packing
perspective. We also show that the orthogonal basis uniquely minimizes the
$p$-frame three-point energy when $0<p<1$ in all dimensions. The arguments make
use of multivariate polynomials employed in semidefinite programming and based
on the classical Gegenbauer polynomials. For $p=1$, we completely solve the
analogous problem on the circle. As for higher dimensions, we show that the
Hausdorff dimension of minimizers is not greater than $d-2$ for measures on
$\mathbb{S}^{d-1}$. As the main ingredient of our proof, we show that the only
isotropic measure without obtuse angles is the uniform distribution over an
orthonormal basis.

Classical Analysis and ODEs;Combinatorics