Small Designs for Path Connected Spaces and Path Connected Homogeneous Spaces

TL;DR

This paper demonstrates the existence of small-sized designs in various contexts, including n-designs on spheres, using novel techniques that improve size bounds.

Contribution

The paper introduces new methods to establish the existence of small designs, notably for n-designs on spheres with improved size bounds.

Findings

Existence of n-designs on spheres with size O_d(n^d log(n)^{d-1})

Techniques applicable to various contexts for small design existence

Improved bounds on the size of designs in geometric spaces

Abstract

We prove the existence of designs of small size in a number of contexts. In particular our techniques can be applied to prove the existence of $n$-designs on $S^{d}$ of size $O_d(n^{d}\log(n)^{d-1})$.