A variational characterisation of spherical designs

In this paper we first establish a new variational characterisation of spherical designs: it is shown that a set XN={x1,…,xN}⊂Sd, where Sd:={x∈Rd+1:∑j=1dxj2=1}, is a spherical LL-design if and only if a certain non-negative quantity AL,N(XN)AL,N(XN) vanishes. By combining this result with a known “sampling theorem” for the sphere, we obtain the main result, which is that if XN⊂SdXN⊂Sd is a stationary point set of AL,NAL,N whose “mesh norm” satisfies hXN<1/(L+1)hXN<1/(L+1), then XNXN is a spherical LL-design. The latter result seems to open a pathway to the elusive problem of proving (for fixed dd) the existence of a spherical LL-design with a number of points NN of order (L+1)d(L+1)d. A numerical example with d=2d=2 and L=19L=19 suggests that computational minimisation of AL,NAL,N can be a valuable tool for the discovery of new spherical designs for moderate and large values of LL.