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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Extremal properties of spherical semidesigns


Authors: N. O. Kotelina and A. B. Pevnyĭ
Translated by: the authors
Original publication: Algebra i Analiz, tom 22 (2010), nomer 5.
Journal: St. Petersburg Math. J. 22 (2011), 795-801
MSC (2010): Primary 52C35
Published electronically: June 28, 2011
MathSciNet review: 2828829
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Abstract: 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.


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Additional Information

N. O. Kotelina
Affiliation: Department of Mathematics, Syktyvkar State University, Syktyvkar 167001, Russia
Email: nad7175@yandex.ru

A. B. Pevnyĭ
Affiliation: Department of Mathematics, Syktyvkar State University, Syktyvkar 167001, Russia
Email: pevnyi@syktsu.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-2011-01168-9
PII: S 1061-0022(2011)01168-9
Keywords: Spherical designs, spherical semidesigns
Received by editor(s): August 4, 2009
Published electronically: June 28, 2011
Article copyright: © Copyright 2011 American Mathematical Society