Reverse Alexandrov-Fenchel and Heintze-Karcher inequalities are proved with deficits controlled by oriented volumes of focal maps, yielding sharp estimates in Minkowski planes, Euclidean space, spheres, and hyperbolic space.
Evolutes and isoperimetric deficit in two-dimensional spaces of constant curvature.Archivum Mathematicum, 50(4):219–236, 2014
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Heintze-Karcher and Reverse Alexandrov-Fenchel Inequalities via Focal Geometry
Reverse Alexandrov-Fenchel and Heintze-Karcher inequalities are proved with deficits controlled by oriented volumes of focal maps, yielding sharp estimates in Minkowski planes, Euclidean space, spheres, and hyperbolic space.