  We show that for every closed Riemannian manifold there exists a continuous
family of $1$-cycles (defined as finite collections of disjoint closed curves)
parametrized by a sphere and sweeping out the whole manifold so that the
lengths of all connected closed curves are bounded in terms of the volume (or
the diameter) and the dimension $n$ of the manifold, when $n \geq 3$. An
alternative form of this result involves a modification of Gromov's definition
of waist of sweepouts, where the space of parameters can be any finite
polyhedron (and not necessarily a pseudomanifold). We demonstrate that the
so-defined polyhedral $1$-dimensional waist of a closed Riemannian manifold is
equal to its filling radius up to at most a constant factor. We also establish
upper bounds for the polyhedral $1$-waist of some homology classes in terms of
the volume or the diameter of the ambient manifold. In addition, we provide
generalizations of these results for sweepouts by polyhedra of higher dimension
using the homological filling functions. Finally, we demonstrate that the
filling radius and the hypersphericity of a closed Riemannian manifold can be
arbitrarily far apart.
