Intrinsic volumes and polar sets in spherical space

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Rigidity results, inverse curvature flows and Alexandrov-Fenchel type inequalities in the sphere (2013, background): “These theorems allow us to solve isoperimetric-type problems in the sphere: Since the isoperimetric problem is well understood in spherical space and the minimizers are known to be geodesic spheres, we can partially answer a question posed in [6], which can be regarded as a generalized isoperimetric”
Living on the edge: phase transitions in convex programs with random data (2013, result): “chneider & Weil [SW08, p. 263] ask what relationships hold among the conic intrinsic volumes vk of a convex cone. To the best of our knowledge, the literature contains only two nontrivial results [GHS02]: Among all cones in Cd with a ﬁxed value of vd, a circular cone minimizes vd¡1 and maximizes”