We prove several rigidity theorems related to and including Lytchak's problem. The focus is on Alexandrov spaces with $operatorname{curv}geq1$, 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, 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.
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