EXTREMAL PROPERTIES OF SPHERICAL SEMIDESIGNS

Abstract: For every even $ tgeq2$ 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
angle]^{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. - See more at: http://www.ams.org/journals/spmj/2011-22-05/S1061-0022-2011-01168-9/#sthash.BXnK353D.dpuf