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 \(n \ge 3\). 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.

我们表明，对于每个封闭的黎曼流形，都存在一个连续的 1 圈族（定义为不相交的封闭曲线的有限集合），由球体参数化并扫除整个流形，因此所有连接的封闭曲线的长度都以项为界体积（或直径）和尺寸的 ñ歧管的，当\（N \ GE 3 \）. 该结果的另一种形式涉及对 Gromov 的扫掠腰部定义的修改，其中参数空间可以是任何有限多面体（不一定是伪流形）。我们证明了封闭黎曼流形的如此定义的多面体一维腰围等于其填充半径，最多为一个常数因子。我们还根据环境流形的体积或直径为某些同源类的多面体 1-腰部建立了上限。此外，我们使用同调填充函数对高维多面体的清扫提供了这些结果的概括。最后，我们证明了封闭黎曼流形的填充半径和超球面度可以任意远。