Spherical designs via Brouwer fixed point theorem

TL;DR

This paper proves the existence of spherical n-designs with a specified number of points on the sphere S^d for sufficiently large N, using the Brouwer fixed point theorem, advancing the understanding of spherical designs.

Contribution

It establishes a new existence result for spherical designs with explicit bounds on the number of points, employing topological fixed point methods.

Findings

Existence of spherical n-designs for N >= c_d * n^{2d*(d+1)/(d+2)}

Explicit bounds depending only on the dimension d

Application of Brouwer fixed point theorem to spherical design problem

Abstract

For each N>=c_d*n^{2d*(d+1)/(d+2)} we prove the existence of a spherical n-design on S^d consisting of N points, where c_d is a constant depending only on $d$.