The displacement energy \(e(A,B)\), for two compact subsets \(A\) and \(B\) of the cotangent bundle \(T^*M\), measures the minimal energy of Hamiltonian isotopies necessary to displace the subset \(B\) in order to become disjoint with \(A\). The authors introduce a pseudo-distance on Tamarkin’s category \(\mathcal{D}(M)\), inspired from the work of [M. Kashiwara and P. Schapira, J. Appl. Comput. Topol. 2, No. 1–2, 83–113 (2018; Zbl 1423.55013)] on interleaving distance for persistence modules in the context of sheaves, and give in terms of it a lower bound for the displacement energy. As a corollary, Tamarkin’s separation theorem [D. Tamarkin, Springer Proc. Math. Stat. 269, 99–223 (2018; Zbl 1416.35019)] follows from this lower bound.
The very clearly and well written article carefully proves all results or gives references for known ones. It finishes with several applications and concrete calculi.