On balanced circuits in uniform rank-three oriented matroids

Abstract

A set of four points $\{p_1,p_2,p_3,p_4\}$ on the sphere is a balanced quadruple if there are four real numbers $s_1,s_2,s_3,s_4$, two positive and two negative, such that $s_1{p}_1+s_2{p}_2+s_3{p}_3+s_4{p}_4=0$. Streltsova and Wagner proved that $n$ points on the sphere determine at least $\frac{1}{4}\lfloor \frac{n}{2}\rfloor \lfloor \frac{n-1}{2}\rfloor \lfloor \frac{n-2}{2}\rfloor \lfloor \frac{n-3}{2}\rfloor$ many balanced quadruples provided any three points are linearly independent. We extend this result from spherical point configurations to rank-three oriented matroids.