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The isoperimetric inequality for a smooth compact Riemannian manifold $A$provides a positive ${\bf c}(A)$, so that for any $k+1$ dimensional integralcurrent $S_0$ in $A$ there exists an integral current $ S$ in $A$ with$\partial S=\partial S_0$ and ${\bf M}(S)\leq {\bf c}(A){\bf M}(\partialS)^{(k+1)/k}$. Although such an inequality still holds for any compactLipschitz neighborhood retract $A$, it may fail in case $A$ contains a singlepolynomial singularity. Here, replacing $(k+1)/k$ by $1$, we find that a linearinequality ${\bf M}(S)\leq {\bf c}(A){\bf M}(\partial S)$ is valid for anycompact 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 withcoefficients in any complete normed abelian group. A relative version for asubanalytic pair $B\subset A$ is also true, and there are applications tovariational and metric properties of subanalytic sets.

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