Extremal properties of spherical semidesigns

Abstract:

For every even $t\geq 2$ and every set of vectors $\Phi =\{\varphi _1,\dots ,\varphi _m\}$ on the sphere $S^{n-1}$, the notion of the $t$-potential $P_{t}(\Phi )=\sum ^{m}_{i,j=1}[\langle \varphi _i,\varphi _j \rangle ]^{t}$ is introduced. It is proved that the minimum value of the $t$-potential is attained at the spherical semidesigns of order $t$ and only at them. The first result of this type was obtained by B. B. Venkov. The result is extended to the case of sets $\Phi$ that do not lie on the sphere $S^{n-1}$. For the V. A. Yudin potentials $U_{k}(\Phi )$, $k=2,4,\dots ,t$, it is shown that they attain the minimal value at the spherical semidesigns of order $t$ and only at them.