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The analysis of the stability and efficiency of algorithms for convex optimization naturally leads to the study of condition numbers. The Grassmann condition, which is a geometric version of Renegar’s condition, is especially suited for a probabilistic analysis. Such analysis can be performed by relying on techniques from spherical convex geometry and differential geometry. Along this way, we obtain an average analysis of the Grassmann condition number that holds for any regular convex cone. A closer look prompts the investigation of the spherical counterparts of intrinsic volumes -- a notion thoroughly studied for euclidean spaces, but much less so for spheres, so that many fascinating questions remain. (Joint work with Dennis Amelunxen)