(Isoperimetric inequality cf. [63, Lem 3.10]) Let $L$ be a closed Lagrangian integral current in $(X,\omega,\Omega)$, and consider a Euclidean coordinate ball $B_{2R}$ on which the regularity scale is at least $O(R)$. Assume the quantitative almost calibrated condition $\cos\theta\geq\sin\epsilon$. Then there is a universal constant $C$ depending only on $\epsilon,n$ and the metric uniform equivalence constant in (55), so that

$$Mass(A)^{(n-1)/n}\leq CMass(\partial A)$$

for all closed subsets $A$ of $\text{supp}(L)\cap B_{R}$ with rectifiable boundary. Here the Hausdorff measures are computed using the Calabi-Yau metric.