# Optimal asymptotic bounds for spherical designs

@article{Bondarenko2010OptimalAB,
title={Optimal asymptotic bounds for spherical designs},
author={Andriy V. Bondarenko and Danylo V. Radchenko and Maryna S. Viazovska},
journal={arXiv: Metric Geometry},
year={2010}
}
• Published 22 September 2010
• Mathematics
• arXiv: Metric Geometry
In this paper we prove the conjecture of Korevaar and Meyers: for each $N\ge c_dt^d$ there exists a spherical $t$-design in the sphere $S^d$ consisting of $N$ points, where $c_d$ is a constant depending only on $d$.
132 Citations

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## References

SHOWING 1-10 OF 25 REFERENCES

### Spherical Designs via Brouwer Fixed Point Theorem

• Mathematics
SIAM J. Discret. Math.
• 2010
It is shown that c_{d}t is a constant depending only on d, and the existence of a spherical design on S^{d} consisting of N points is proved.

### Lower bounds for spherical designs

A?new lower bound is obtained for the cardinality of spherical designs. In its dependence on dimension it improves the well-known bound of Delsarte by an exponential factor as the degree of the

### On tight spherical designs

• Mathematics
• 2012
Let X be a tight t-design of dimension n for one of the open cases t=5 or t=7. An investigation of the lattice generated by X using arithmetic theory of quadratic forms allows to exclude infinitely

### Nonexistence of Certain Spherical Designs of Odd Strengths and Cardinalities

• Mathematics
Discret. Comput. Geom.
• 1999
This paper obtains some necessary conditions for the existence of designs of odd strengths and cardinalities, and derives a bound which is better than the corresponding estimation ensured by the Delsarte—Goethals—Seidel bound.

### Distribution of points on spheres and approximation by zonotopes

• Mathematics
• 1988
It is proved that if we approximate the Euclidean ballBn in the Hausdorff distance up toɛ by a Minkowski sum ofN segments, then the smallest possibleN is equal (up to a possible logarithmic factor)

### Well Conditioned Spherical Designs for Integration and Interpolation on the Two-Sphere

• Mathematics
SIAM J. Numer. Anal.
• 2010
This paper shows how to construct well conditioned spherical designs with $N\geq(t+1)^2$ points by maximizing the determinant of a matrix while satisfying a system of nonlinear constraints.

### Construction of spherical t-designs

Spherical t-designs are Chebyshev-type averaging sets on the d-dimensional unit sphere Sd−1, that are exact for polynomials of degree at most t. The concept of such designs was introduced by

### Existence of Solutions to Systems of Underdetermined Equations and Spherical Designs

• Mathematics
SIAM J. Numer. Anal.
• 2006
It is shown that the construction of spherical designs is equivalent to solution of underdetermined equations and a new verification method for underd determined equations is derived using Brouwer’s fixed point theorem.

### Computational existence proofs for spherical t-designs

• Computer Science, Mathematics
Numerische Mathematik
• 2011
A computational algorithm based on interval arithmetic which, for given t, upon successful completion will have proved the existence of a t-design with (t + 1)2 nodes on the unit sphere and will have computed narrow interval enclosures which are known to contain these nodes with mathematical certainty.

### Sphere packings, I

• T. Hales
• Physics, Mathematics
Discret. Comput. Geom.
• 1997
A program to prove the Kepler conjecture on sphere packings is described and it is shown that every Delaunay star that satisfies a certain regularity condition satisfies the conjecture.