On the sharp lower bound for duality of modulus

We establish a sharp reciprocity inequality for modulus in compact metric spaces X X with finite Hausdorff measure. In particular, when X X is also homeomorphic to a planar rectangle, our result answers a question of K. Rajala and M. Romney [Ann. Acad. Sci. Fenn. Math. 44 (2019), pp. 681-692]. More specifically, we obtain a sharp inequality between the modulus of the family of curves connecting two disjoint continua E E and F F in X X and the modulus of the family of surfaces of finite Hausdorff measure that separate E E and F F . The paper also develops approximation techniques, which may be of independent interest.