2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/146566If F is a family of mod 2 flat k-cycles in the unit n-ball, we lower bound the maximal volume of any cycle in F in terms of the homology class of F in the space of all cycles. We give examples to show that these lower bounds are fairly sharp.55 pages, 9 figures. The results about cup powers are due to Gromov. When I wrote the first version of the paper I believed that they were new. They are now attributed correctly. Also, the exposition is improved. Accepted for publication in Geometric and Functional AnalysisDifferential Geometry53C23text