Heintze-Karcher and Reverse Alexandrov-Fenchel Inequalities via Focal Geometry

Abstract

We prove a collection of reverse Alexandrov-Fenchel type inequalities in anisotropic, Euclidean, spherical, and hyperbolic settings. The unifying principle is that the relevant deficit is controlled by curvature radius data, or equivalently by the signed volume of an associated evolute or focal map. For smooth simple strictly convex curves in a smooth Minkowski plane we prove an anisotropic Hurwitz-type inequality: the anisotropic isoperimetric deficit is bounded above by the signed Euclidean area of the Minkowski evolute. For smooth closed strictly convex hypersurfaces in $\mathbb R^{n+1}$, with $E_k$ denoting the normalised $k$-th mean curvature, we establish the sharp reverse Alexandrov-Fenchel estimate \[ 0\le \frac{1}{|\mathbb S^{n}|} \left(\int_{M}E_{n-1}\,d\mu\right)^{2} -\int_{M}E_{n-2}\,d\mu \le \frac{n}{2(n+1)} \int_{M}\frac{E_{n-1}^{2}-E_{n-2}E_{n}}{E_{n}}\,d\mu . \] We also relate the deficit of the Minkowski inequality to the oriented volumes of the two focal maps. In space forms we derive a normal-graph formula for oriented volume and use it to give focal-map interpretations of the deficit of an unweighted Heintze--Karcher inequality. In dimension two this recovers evolute-area formulae. We then prove exact reverse isoperimetric identities for curves on $\mathbb S^{2}$ and strictly horoconvex curves in $\mathbb H^{2}$, in which the remainders are explicit nonnegative integrals measuring the oscillation of geodesic curvature. In the spherical case, for every smooth simple closed curve $\gamma\subset\mathbb S^2$ with length $L$ and enclosed area $A$, if $k_E$ denotes the ambient Euclidean curvature of $\gamma$, then \[ L^{2}-A(4\pi-A) \le \left(\int_{\gamma}k_E\,ds\right)^{2}-4\pi^{2}, \] with equality if and only if $\gamma$ is a geodesic circle. This relates the spherical isoperimetric deficit with the Euclidean Fenchel deficit.