Strong rotundity and optimization

Standard techniques from the study of well-posedness show that if a fixed convex objective function is minimized in turn over a sequence of convex feasible regions converging Mosco to a limiting feasible region, then the optimal solutions converge in norm to the optimal solution of the limiting problem. Certain conditions on the objective function are needed as is a constraint qualification. If, as may easily occur in practice, the constraint qualification fails, stronger set convergence is required, together with stronger analytic/geometric properties of the objective function: strict convexity (to ensure uniqueness), weakly compact level sets (to ensure existence and weak convergence), and the Kadec property (to deduce norm convergence). By analogy with the L_p norms, such properties are termed "strong rotundity." A very simple characterization of strongly rotund integral functionals on L₁ is presented that shows, for example, that the Boltzmann-Shannon entropy ∫ x log x is strongly rotund. Examples are discussed, and the existence of everywhere- and densely-defined strongly rotund functions is investigated.