The authors introduce new metrics on spaces of Lagrangian submanifolds, not necessarily in a fixed Hamiltonian isotopy class. These metrics arise from measurements involving Lagrangian cobordisms. They also show that splitting Lagrangians through cobordism has an energy cost and, from this cost being smaller than certain explicit bounds, they deduce some forms of rigidity of Lagrangian intersections. The authors also fit these constructions in the more general algebraic setting of triangulated categories, independent of Lagrangian cobordisms and, as a main technical tool, they develop aspects of the theory of (weakly) filtered \(A_{\infty}-\)categories.