We will discuss parallels between the problems of minimizing the total mass vs. maximizing the boundary capacity for asymptotically flat manifolds with nonnegative scalar curvature, subject to natural boundary conditions. The former is well known as Bartnik's approach to quasi-local mass, whereas the latter is apparently new. For minimizing (maximizing) a functional, one seeks lower (upper) semicontinuity. Consequently, we will discuss joint work with Dan Lee on the lower semicontinuity of ADM mass and with Raquel Perales and Jim Portegies on the upper semicontinuity of capacity, all for Sormani--Wenger intrinsic flat convergence.