  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.
