Eilenberg's inequality

A spherical design, part of combinatorial design theory in mathematics, is a finite set of N points on the d-dimensional unit hypersphere S^d such that the average value of any polynomial f of degree t or less on the set equals the average value of f on the whole sphere (that is, the integral of f over S^d divided by the area or measure of S^d). Such a set is often called a spherical t-design to indicate the value of t, which is a fundamental parameter. Spherical t-designs for different values of N and t can be found precomputed at http://www.research.att.com/~njas/sphdesigns.

Spherical designs can be of value in approximation theory, in statistics for experimental design (being usable to construct rotatable designs), in combinatorics, and in geometry. The main problem is to find examples, given d and t, that are not too large. However, such examples may be hard to come by. Spherical t-designs have also recently been appropriated in quantum mechanics in the form of quantum t-designs with various applications to quantum information theory, quantum computing and POVMs.

The concept of a spherical design is due to Delsarte, Goethals, and Seidel (1977). The existence and structure of spherical designs with d = 1 (that is, in a circle) was studied in depth by Hong (1982). Shortly thereafter, Seymour and Zaslavsky (1984) proved that such designs exist of all sufficiently large sizes; that is, there is a number N(d,t) such that for every N ≥ N(d,t) there exists a spherical t-design of N points in dimension d. However, their proof gave no idea of how big N(d,t) is. Good estimates for that were found later on. Besides these "large" sizes, there are many sporadic small spherical designs; many of them are related to finite group actions on the sphere and are of great interest in themselves.

Recently, Bondarenko, Radchenko, and Viazovska obtained the optimal asymptotic upper bound $N(d,t)<C_d{t}^d$ for all positive integers d and t.

One application of spherical designs is for whole-sphere data collection. Spherical t-designs meet the "accurately approximate integrals by sums" criteria for "good" pixelizations of the sphere.

See also

Thomson problem

References

• Bondarenko, A., Radchenko D., and Viazovska M. (2013), "On optimal asymptotic bounds for spherical designs", Annals of Mathematics, vol. 178, pp. 443–452.
• Delsarte, P., Goethals, J.M., and Seidel, J.J. (1977), "Spherical codes and designs." Geometriae Dedicata vol. 6, pp. 363–388.
• Hong, Yiming (1982), "On spherical t-designs in R²." European Journal of Combinatorics, vol. 3, pp. 255–258.
• Seidel, J.J. (1991), Geometry and Combinatorics: Selected Works of J.J. Seidel. D.G. Corneil and R. Mathon, eds. Boston: Academic Press. Reprints Delsarte et al. (1977).
• Seymour, P.D., and Zaslavsky, Thomas (1984), "Averaging sets: A generalization of mean values and spherical designs." Advances in Mathematics, vol. 52, pp. 213–240. Much more general than spheres.

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