From point sets to curves: 𝑡-Designs and Marcinkiewicz-Zygmund inequalities on the sphere

A geodesic cycle is a closed curve that connects finitely many points along geodesics. We study geodesic cycles on the sphere in regard to their role in equal-weight quadrature via t t -designs and in approximation theory through Marcinkiewicz-Zygmund inequalities. In the first part we analyze and construct explicit geodesic cycles that lead to t t -design curves for small t t . In the second part we prove the existence of geodesic cycles satisfying Marcinkiewicz-Zygmund inequalities with asymptotically optimal length.