  Over the last years, minimization problems over spaces of measures have
received increased interest due to their relevance in the context of inverse
problems, optimal control and machine learning. A fundamental role in their
numerical analysis is played by the assumption that the optimal dual state
admits finitely many global extrema and satisfies a second-order sufficient
optimality condition in each one of them. In this work, we show the full
equivalence of these structural assumptions to a no-gap second-order condition
involving the second subderivative of the Radon norm as well as to a local
quadratic growth property of the objective functional with respect to the
bounded Lipschitz norm.
