We use a variational formulation to define a generalized notion of perimeter, from which we derive abstract isoperimetric Cheeger's inequalities via gradient estimates on solutions of Poisson equations. Our abstract framework unifies many existing results and also allows us to prove a new transport inequality, which strengthens the already known transport-information inequality. Conversely, we also prove that Cheeger's inequality implies certain Calderón-Zygmund gradient estimates.