Linear isoperimetric inequality for normal and integral currents in compact subanalytic sets

Abstract

The isoperimetric inequality for a smooth compact Riemannian manifold $A$ provides a positive ${\bf c}(A)$, so that for any $k+1$ dimensional integral current $S_0$ in $A$ there exists an integral current $ S$ in $A$ with $\partial S=\partial S_0$ and ${\bf M}(S)\leq {\bf c}(A){\bf M}(\partial S)^{(k+1)/k}$. Although such an inequality still holds for any compact Lipschitz neighborhood retract $A$, it may fail in case $A$ contains a single polynomial singularity. Here, replacing $(k+1)/k$ by $1$, we find that a linear inequality ${\bf M}(S)\leq {\bf c}(A){\bf M}(\partial S)$ is valid for any compact algebraic, semi-algebraic, or even subanalytic set $A$. In such a set, this linear inequality holds not only for integral currents, which have $\boldsymbol{Z}$ coefficients, but also for normal currents having $\boldsymbol{R}$ coefficients and generally for normal flat chains with coefficients in any complete normed abelian group. A relative version for a subanalytic pair $B\subset A$ is also true, and there are applications to variational and metric properties of subanalytic sets.