# Commentarii Mathematici Helvetici

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**Volume 86, Issue 3, 2011, pp. 557–591**

**DOI: 10.4171/CMH/234**

Flat currents modulo `p` in metric spaces and filling radius inequalities

^{[1]}and Mikhail G. Katz

^{[2]}(1) Classe di Scienze M F N, Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126, PISA, ITALY

(2) Department of Mathematics and Statistics, Bar-Ilan University, 52 900, RAMAT-GAN, ISRAEL

We adapt the theory of currents in metric spaces, as developed by the
first-mentioned author in collaboration with B. Kirchheim, to currents
with coefficients in ℤ_{p} . We obtain isoperimetric
inequalities mod (`p`) in Banach spaces and we apply these inequalities
to provide a proof of Gromov’s filling radius inequality which applies
also to nonorientable manifolds. With this goal in mind, we use the
Ekeland principle to provide quasi-minimizers of the mass mod (`p`) in
the homology class, and use the isoperimetric inequality to give lower
bounds on the growth of their mass in balls.

*Keywords: *Filling radius, currents, isoperimetric inequality

Ambrosio Luigi, Katz Mikhail: Flat currents modulo `p` in metric spaces and filling radius inequalities. *Comment. Math. Helv.* 86 (2011), 557-591. doi: 10.4171/CMH/234