On Stegall's smooth variational principle (Q858632)

A well-known perturbation of the result in the title is that if a Banach space has the Radon--Nikodým Property, then any coercive lower semicontinuous bounded below function defined on it admits an arbitrarily small linear variation which attains a strong minimum. The authors note that the converse is also true. More significantly, they prove that either of the following formally weaker conditions is also equivalent to the Radon--Nikodým Property: (i) any coercive continuous bounded below function defined on the space admits a linear variation which attains a minimum, (ii) any coercive continuous convex bounded below function defined on the space admits a linear variation which attains a strong minimum. One may note that the obvious common weakening of these two conditions is not interesting, as it holds for every Banach space.