  Abstract. In this paper we prove several rigidity theorems related to and
including Lytchak's problem. The focus is on Alexandrov spaces with \curv\geq1,
nonempty boundary, and maximal radius \frac{\pi}{2}. We exhibit many such
spaces that indicate that this class is remarkably flexible. Nevertheless, we
also show that when the boundary is either geometrically or topologically
spherical, then it is possible to obtain strong rigidity results. In contrast
to this one can show that with general lower curvature bounds and strictly
convex boundary only cones can have maximal radius. We also mention some
connections between our problems and the positive mass conjectures. This paper
is an expanded version and replacement of the two previous versions
