Alexandrov-Fenchel Type Inequalities Research Articles

Stability of Alexandrov–Fenchel Type Inequalities for Nearly Spherical Sets in Space Forms

Stability of Alexandrov–Fenchel Type Inequalities for Nearly Spherical Sets in Space Forms

Anisotropic Alexandrov–Fenchel Type Inequalities and Hsiung–Minkowski Formula

Anisotropic Alexandrov–Fenchel Type Inequalities and Hsiung–Minkowski Formula

Weighted Alexandrov–Fenchel type inequalities for hypersurfaces in Rn$\mathbb {R}^n$

AbstractIn this paper, we prove the following geometric inequalities in the Euclidean space , which are weighted Alexandrov–Fenchel type inequalities, provided that is a star‐shaped and ‐convex hypersurface. Equality holds if and only if is a coordinate sphere in . As an application, by letting in the above inequality, we obtain a lower bound for the outer radius in terms of the curvature integrals for star‐shaped and ‐convex hypersurfaces.

The Alexandrov–Fenchel type inequalities, revisited

The Alexandrov–Fenchel type inequalities, revisited

Alexandrov-Fenchel type inequalities in the sphere

In this paper, we proved the Alexandrov-Fenchel inequalities for embedded, closed, connected and convex C2-hypersurfaces in Sn+1: Ak≥ξk,k−2(Ak−2) for any 1≤k≤n−1, where Ak is the quermassintegral (see Definition 1.1) and ξk,k−2 is the unique positive function such that the equality holds when M is a geodesic sphere.

Volume preserving flow and Alexandrov–Fenchel type inequalities in hyperbolic space

In this paper, we study flows of hypersurfaces in hyperbolic space, and apply them to prove geometric inequalities. In the first part of the paper, we consider volume preserving flows by a family of curvature functions including positive powers of k -th mean curvatures with k=1,\ldots,n , and positive powers of p -th power sums S_p with p > 0 . We prove that if the initial hypersurface M_0 is smooth and closed and has positive sectional curvatures, then the solution Mt of the flow has positive sectional curvature for any time t > 0 , exists for all time and converges to a geodesic sphere exponentially in the smooth topology. The convergence result can be used to show that certain Alexandrov–Fenchel quermassintegral inequalities, known previously for horospherically convex hypersurfaces, also hold under the weaker condition of positive sectional curvature. In the second part of this paper, we study curvature flows for strictly horospherically convex hypersurfaces in hyperbolic space with speed given by a smooth, symmetric, increasing and degree one homogeneous function f of the shifted principal curvatures \lambda_i=\kappa_i-1 , plus a global term chosen to impose a constraint on the quermassintegrals of the enclosed domain, where f is assumed to satisfy a certain condition on the second derivatives. We prove that if the initial hypersurface is smooth, closed and strictly horospherically convex, then the solution of the flow exists for all time and converges to a geodesic sphere exponentially in the smooth topology. As applications of the convergence result, we prove a new rigidity theorem on smooth closed Weingarten hypersurfaces in hyperbolic space, and a new class of Alexandrov–Fenchel type inequalities for smooth horospherically convex hypersurfaces in hyperbolic space.

Locally constrained curvature flows and geometric inequalities in hyperbolic space

In this paper, we first study the locally constrained curvature flow of hypersurfaces in hyperbolic space, which was introduced by Brendle, Guan and Li [7]. This flow preserves the $m$th quermassintegral and decreases $(m+1)$th quermassintegral, so the convergence of the flow yields sharp Alexandrov-Fenchel type inequalities in hyperbolic space. Some special cases have been studied in [7]. In the first part of this paper, we show that h-convexity of the hypersurface is preserved along the flow and then the smooth convergence of the flow for h-convex hypersurfaces follows. We then apply this result to establish some new sharp geometric inequalities comparing the integral of $k$th Gauss-Bonnet curvature of a smooth h-convex hypersurface to its $m$th quermassintegral (for $0\leq m\leq 2k+1\leq n$), and comparing the weighted integral of $k$th mean curvature to its $m$th quermassintegral (for $0\leq m\leq k\leq n$). In particular, we give an affirmative answer to a conjecture proposed by Ge, Wang and Wu in 2015. In the second part of this paper, we introduce a new locally constrained curvature flow using the shifted principal curvatures. This is natural in the context of h-convexity. We prove the smooth convergence to a geodesic sphere of the flow for h-convex hypersurfaces, and provide a new proof of the geometric inequalities proved by Andrews, Chen and the third author of this paper in 2018. We also prove a family of new sharp inequalities involving the weighted integral of $k$th shifted mean curvature for h-convex hypersurfaces, which as application implies a higher order analogue of Brendle, Hung and Wang's [8] inequality.

Harmonic mean curvature flow and geometric inequalities

We employ the harmonic mean curvature flow of strictly convex closed hypersurfaces in hyperbolic space to prove Alexandrov-Fenchel type inequalities relating quermassintegrals to the total curvature, which is the integral of Gaussian curvature on the hypersurface. The resulting inequality allows us to use the inverse mean curvature flow to prove Alexandrov-Fenchel inequalities between the total curvature and the area for strictly convex hypersurfaces. Finally, we apply the harmonic mean curvature flow to prove a new class of geometric inequalities for h-convex hypersurfaces in hyperbolic space.

Dual Orlicz–Brunn–Minkowski theory: Dual Orlicz Lϕ affine and geominimal surface areas

This paper aims to develop the basic theory for the dual Orlicz Lϕ affine and geominimal surface areas of star bodies. Basic properties of these new affine invariants are established. Moreover, related Orlicz affine isoperimetric, cyclic, Santaló type and Alexandrov–Fenchel type inequalities are obtained.

Inequalities of Alexandrov–Fenchel type for convex hypersurfaces in hyperbolic space and in the sphere

In this paper, firstly, inspired by Nat\'{a}rio's recent work \cite{Na}, we use the isoperimetric inequality to derive some Alexandrov-Fenchel type inequalities for closed convex hypersurfaces in the hyperbolic space $\H^{n+1}$ and in the sphere $\SS^{n+1}$. We also get the rigidity in the spherical case. Secondly, we use the inverse mean curvature flow in sphere \cite{gerh,Mak-Sch} to prove an optimal Sobolev type inequality for closed convex hypersurfaces in the sphere.

Mixedf-divergence and inequalities for log-concave functions

Mixed f-divergences, a concept from information theory and statistics, measure the difference between multiple pairs of distributions. We introduce them for log-concave functions and establish some of their properties. Among them are affine invariant vector entropy inequalities, like new Alexandrov–Fenchel-type inequalities and an affine isoperimetric inequality for the vector form of the Kullback Leibler divergence for log-concave functions. Special cases of f-divergences are mixed L λ -affine surface areas for log-concave functions. For those, we establish various affine isoperimetric inequalities as well as a vector Blaschke Santaló-type inequality.

Isoperimetric type problems and Alexandrov–Fenchel type inequalities in the hyperbolic space

In this paper, we solve various isoperimetric problems for the quermassintegrals and the curvature integrals in the hyperbolic space Hn, by using quermassintegral preserving curvature flows. As a byproduct, we obtain hyperbolic Alexandrov–Fenchel inequalities.

Inequalities for general mixed affine surface areas

Several general mixed affine surface areas are introduced. We prove some important properties, such as, affine invariance, for these general mixed affine surface areas. We also establish new Alexandrov-Fenchel type inequalities, Santal\'{o}-type inequalities, and affine isoperimetric inequalities for these general mixed affine surface areas.

Inequalities for mixed p-affine surface area

We prove new Alexandrov-Fenchel type inequalities and new affine isoperimetric inequalities for mixed $p$-affine surface areas. We introduce a new class of bodies, the illumination surface bodies, and establish some of their properties. We show, for instance, that they are not necessarily convex. We give geometric interpretations of $L_p$ affine surface areas, mixed $p$-affine surface areas and other functionals via these bodies. The surprising new element is that not necessarily convex bodies provide the tool for these interpretations.

Rigidity results, inverse curvature flows and Alexandrov–Fenchel type inequalities in the sphere

We prove a rigidity result in the sphere which allows us to generalize a result about smooth convex hypersurfaces in the sphere by Do Carmo-Warner to convex $C^2$-hypersurfaces. We apply these results to prove $C^{1,\beta}$-convergence of inverse F-curvature flows in the sphere to an equator in \mathbb{S}^{n+1} for embedded, closed, strictly convex initial hypersurfaces. The result holds for large classes of curvature functions including the mean curvature and arbitrary powers of the Gauss curvature. We use this result to prove Alexandrov-Fenchel type inequalities in the sphere.