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.
The geometry of Minkowski spaces-a survey
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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.