Extremal properties of spherical semidesigns

For every even $t\geq2$ 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.