Geodesic
Extremal properties of spherical semidesigns
Algebra i analiz, Tome 22 (2010) no. 5, pp. 131-139.

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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.
Keywords: spherical designs, spherical semidesigns. 
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N. O. Kotelina; A. B. Pevnyǐ. Extremal properties of spherical semidesigns. Algebra i analiz, Tome 22 (2010) no. 5, pp. 131-139. http://geodesic.mathdoc.fr/item/AA_2010_22_5_a4/

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