Spherical designs via Brouwer fixed point theorem
arXiv:0811.141616 citationsh-index: 14
Originality Incremental advance
AI Analysis
This provides an improved existence bound for spherical designs, which are important in approximation theory and numerical integration.
The authors prove the existence of spherical n-designs on S^d with N points for N at least c_d * n^{2d(d+1)/(d+2)}, improving previous bounds. The result is obtained via the Brouwer fixed point theorem.
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 $d$.