2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/148800Let X be a closed manifold of dimension 2m >= 6 with torsion-free middle-dimensional homology. We construct metrics on X of arbitrarily small volume, such that every middle-dimensional submanifold of less than unit volume necessarily bounds. Thus, Loewner's theorem has no higher-dimensional analogue.LaTeX2e, 11 pages. Revised version, to appear in Math. Research LettersDifferential GeometryAlgebraic Topology53C23 (Primary) 55Q15 (Secondary)text