  In this paper we extend the results of "A strong minimax property of
nondegenerate minimal submanifolds" by White, where it is proved that any
smooth, compact submanifold, which is a strictly stable critical point for an
elliptic parametric functional, is the unique minimizer in a certain geodesic
tubular neighbourhood. We prove a similar result, replacing the tubular
neighbourhood with one induced by the flat distance and we provide quantitative
estimates. Our proof is based on the introduction of a penalized minimization
problem, in the spirit of "A selection principle for the sharp quantitative
isoperimetric inequality" by Cicalese and Leonardi, which allows us to exploit
the regularity theory for almost minimizers of elliptic parametric integrands.
