Positive cubature on $S^2$:low-degree rigidity and uniform bounds

Source abstract

Let $N_t$ denote the least number of nodes in a positive cubature formula of degree $t$ on $S^2$. We prove that a formula of degree $2m+1$ cannot have exactly $(m+1)(m+2)+1$ nodes for any $m\ge2$. Excluding equality in the Fisher bound then gives $$N_{2m+1}\ge (m+1)(m+2)+2$$ for $m\ge3$, and known constructions yield the exact values $N_7=22$ and $N_9=32$. For general odd degrees, positive circle measures on Lobatto latitudes give an upper bound with quadratic coefficient $13/8$, parity-dependent linear terms, and an $O(m^{2/3})$ remainder. We determine the sharp constant $49\sqrt[3]{3}/72$ for the scalar remainder in this construction. Lower bounds are obtained from continuous weighted LP--Turán inequalities and radial caps whose Helmholtz companions are nonnegative measures. We prove that the cap functional admits a maximizer at each fixed admissible support radius and derive explicit finite-degree lower bounds by a positivity-preserving transfer to the sphere.