On the spread of positively curved Alexandrov spaces

It was proved by F. Wilhelm that Gromov’s filling radius of closed positively curved manifolds with a uniform lower bound on sectional curvature attains the maximum with the round sphere. Recently the author proved that this is also the case for closed finite-dimensional Alexandrov spaces with a positive lower curvature bound. These were proved as a corollary of a comparison theorem for the invariant called spread of those spaces. In this paper, we extend the latter result to infinite-dimensional Alexandrov spaces.