Alexandrov-Fenchel Inequalities Research Articles

Michael-Simon type inequalities in hyperbolic space H n + 1 ${\mathbb{H}}^{n+1}$ via Brendle-Guan-Li’s flows

Abstract In the present paper, we first establish and verify a new sharp hyperbolic version of the Michael-Simon inequality for mean curvatures in hyperbolic space H n + 1 ${\mathbb{H}}^{n+1}$ based on the locally constrained inverse curvature flow introduced by Brendle, Guan and Li (“An inverse curvature type hypersurface flow in H n + 1 ${\mathbb{H}}^{n+1}$ ,” (Preprint)) as follows (0.1) ∫ M λ ′ f 2 E 1 2 + | ∇ M f | 2 − ∫ M ∇ ̄ f λ ′ , ν + ∫ ∂ M f ≥ ω n 1 n ∫ M f n n − 1 n − 1 n $$\underset{M}{\int }{\lambda }^{\prime }\sqrt{{f}^{2}{E}_{1}^{2}+\vert {\nabla }^{M}f{\vert }^{2}}-\underset{M}{\int }\langle \bar{\nabla }\left(f{\lambda }^{\prime }\right),\nu \rangle +\underset{\partial M}{\int }f\ge {\omega }_{n}^{\frac{1}{n}}{\left(\underset{M}{\int }{f}^{\frac{n}{n-1}}\right)}^{\frac{n-1}{n}}$$ provided that M is h-convex and f is a positive smooth function, where λ′(r) = coshr. In particular, when f is of constant, (0.1) coincides with the Minkowski type inequality stated by Brendle, Hung, and Wang in (“A Minkowski inequality for hypersurfaces in the anti-de Sitter-Schwarzschild manifold,” Commun. Pure Appl. Math., vol. 69, no. 1, pp. 124–144, 2016). Further, we also establish and confirm a new sharp Michael-Simon inequality for the kth mean curvatures in H n + 1 ${\mathbb{H}}^{n+1}$ by virtue of the Brendle-Guan-Li’s flow (“An inverse curvature type hypersurface flow in H n + 1 ${\mathbb{H}}^{n+1}$ ,” (Preprint)) as below (0.2) ∫ M λ ′ f 2 E k 2 + | ∇ M f | 2 E k − 1 2 − ∫ M ∇ ̄ f λ ′ , ν ⋅ E k − 1 + ∫ ∂ M f ⋅ E k − 1 ≥ p k ◦ q 1 − 1 ( W 1 ( Ω ) ) 1 n − k + 1 ∫ M f n − k + 1 n − k ⋅ E k − 1 n − k n − k + 1 \begin{align}\hfill & \underset{M}{\int }{\lambda }^{\prime }\sqrt{{f}^{2}{E}_{k}^{2}+\vert {\nabla }^{M}f{\vert }^{2}{E}_{k-1}^{2}}-\underset{M}{\int }\langle \bar{\nabla }\left(f{\lambda }^{\prime }\right),\nu \rangle \cdot {E}_{k-1}+\underset{\partial M}{\int }f\cdot {E}_{k-1}\hfill \\ \hfill & \quad \ge {\left({p}_{k}{\circ}{q}_{1}^{-1}\left({W}_{1}\left({\Omega}\right)\right)\right)}^{\frac{1}{n-k+1}}{\left(\underset{M}{\int }{f}^{\frac{n-k+1}{n-k}}\cdot {E}_{k-1}\right)}^{\frac{n-k}{n-k+1}}\hfill \end{align} provided that M is h-convex and Ω is the domain enclosed by M, p k (r) = ω n (λ′) k−1, W 1 ( Ω ) = 1 n | M | ${W}_{1}\left({\Omega}\right)=\frac{1}{n}\vert M\vert $ , λ′(r) = coshr, q 1 ( r ) = W 1 S r n + 1 ${q}_{1}\left(r\right)={W}_{1}\left({S}_{r}^{n+1}\right)$ , the area for a geodesic sphere of radius r, and q 1 − 1 ${q}_{1}^{-1}$ is the inverse function of q 1. In particular, when f is of constant and k is odd, (0.2) is exactly the weighted Alexandrov–Fenchel inequalities proven by Hu, Li, and Wei in (“Locally constrained curvature flows and geometric inequalities in hyperbolic space,” Math. Ann., vol. 382, nos. 3–4, pp. 1425–1474, 2022).

A complete family of Alexandrov–Fenchel inequalities for convex capillary hypersurfaces in the half-space

A complete family of Alexandrov–Fenchel inequalities for convex capillary hypersurfaces in the half-space

On the volume of the Minkowski sum of zonoids

We explore some inequalities in convex geometry restricted to the class of zonoids. We show the equivalence, in the class of zonoids, between a local Alexandrov-Fenchel inequality, a local Loomis-Whitney inequality, the log-submodularity of volume, and the Dembo-Cover-Thomas conjecture on the monotonicity of the ratio of volume to the surface area. In addition to these equivalences, we confirm these conjectures in R3 and we establish an improved inequality in R2. Along the way, we give a negative answer to a question of Adam Marcus regarding the roots of the Steiner polynomial of zonoids. We also investigate analogous questions in the Lp-Brunn-Minkowski theory, and in particular, we confirm all of the above conjectures in the case p=2, in any dimension.

Uniqueness of solutions to a class of isotropic curvature problems

Employing a local version of the Brunn-Minkowski inequality, we give a new and simple proof of a result due to Andrews, Choi and Daskalopoulos that the origin-centred balls are the only closed, self-similar solutions of the Gauss curvature flow. Extensions to various nonlinearities are obtained, assuming the centroid of the enclosed convex body is at the origin. By applying our method to the Alexandrov-Fenchel inequality, we also show that origin-centred balls are the only solutions to a large class of even Christoffel-Minkowski type problems.

Quantitative quermassintegral inequalities for nearly spherical sets

In this paper, we establish quantitative Alexandrov–Fenchel inequalities for quermassintegrals on nearly spherical sets. In particular, we bound the [Formula: see text]-isoperimetric deficit from below by the Fraenkel asymmetry. We also find a lower bound on the [Formula: see text]-isoperimetric deficit using the spherical deviation.

Locally constrained inverse curvature flow and an Alexandrov-Fenchel inequality in de Sitter space

The long-time existence and convergence are shown for the locally constrained inverse curvature flow of compact spacelike convex hypersurfaces in de Sitter space. As an application, the Alexandrov-Fenchel inequality between the quermassintegrals A0 and An−1 is established in view of their opposite monotonicity along the considered flow. With the help of the dual relationship, a geometric inequality concerning quermassintegrals of convex hypersurfaces in a hyperbolic space of even dimension follows.

A Constrained Mean Curvature Flow and Alexandrov–Fenchel Inequalities

Abstract In this article, we study a locally constrained mean curvature flow for star-shaped hypersurfaces with capillary boundary in the half-space. We prove its long-time existence and the global convergence to a spherical cap. Furthermore, the capillary quermassintegrals defined in [29] evolve monotonically along the flow, and hence we establish a class of new Alexandrov–Fenchel inequalities for convex hypersurfaces with capillary boundary in the half-space.

Alexandrov–Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary

In this paper, we first introduce quermassintegrals for capillary hypersurfaces in the half-space. Then we solve the related isoperimetric type problems for the convex capillary hypersurfaces and obtain the corresponding Alexandrov–Fenchel inequalities. In order to prove these results, we construct a new locally constrained curvature flow and prove that the flow converges globally to a spherical cap.

The extremals of the Alexandrov–Fenchel inequality for convex polytopes

The extremals of the Alexandrov–Fenchel inequality for convex polytopes

Introduction to the combinatorial atlas

We give elementary self-contained proofs of the strong Mason conjecture recently proved by Anari et al. (2018) and Brändén and Huh (2020), and of the classical Alexandrov–Fenchel inequality. Both proofs use the combinatorial atlas technology recently introduced by the authors Chan and Pak (2021). We also give a formal relationship between combinatorial atlases and Lorentzian polynomials.

The relative Alexandrov-Fenchel type inequalities

The Alexandrov-Fenchel inequality is one of the important inequalities in convex geometry, and its special form can be regarded as an isoperimetric inequality between higher-order quermassintegrals. In recent years, hypersurfaces with free or capillary boundaries have received much attention. In this paper, we will study a class of relative Alexandrov-Fenchel-type inequalities on convex hypersurfaces with free or capillary boundaries in the unit ball or in half-space from the perspective of differential geometry, and review its recent progress. First, we introduce the higher-order Minkowski-type integral formula for capillary hypersurfaces in the unit ball or half-space. From the first variational point of view, we give the definition of the quermassintegrals of the capillary hypersurfaces in the unit ball or half-space. After that, we use the Minkowski-type formula to construct a locally constrained curvature flow. By analyzing the long-time existence and convergence of the flow, we prove a class of relative Alexandrov-Fenchel-type inequalities in the unit ball and half-space.

Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in a ball

In this paper, we first introduce the quermassintegrals for convex hypersurfaces with capillary boundary in the unit Euclidean ball B n + 1 {\mathbb {B}}^{n+1} and derive its first variational formula. Then by using a locally constrained nonlinear curvature flow, which preserves the n n -th quermassintegral and non-decreases the k k -th quermassintegral, we obtain the Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in B n + 1 {\mathbb {B}}^{n+1} . This generalizes the result of Scheuer [J. Differential Geom. 120 (2022), pp. 345–373] for convex hypersurfaces with free boundary in B n + 1 {\mathbb {B}}^{n+1} .

Hyperbolic geometry of shapes of convex bodies

We use the intrinsic area to define a distance on the space of homothety classes of convex bodies in the $n$-dimensional Euclidean space, which makes it isometric to a convex subset of the infinite dimensional hyperbolic space. The ambient Lorentzian structure is an extension of the intrinsic area form of convex bodies, and Alexandrov–Fenchel inequality is interpreted as the Lorentzian reversed Cauchy–Schwarz inequality. We deduce that the space of similarity classes of convex bodies has a proper geodesic distance with curvature bounded from below by $-1$ (in the sense of Alexandrov). In dimension 3, this space is homeomorphic to the space of distances with non-negative curvature on the 2-sphere, and this latter space contains the space of flat metrics on the 2-sphere considered by W. P. Thurston. Both Thurston’s and the area distances rely on the area form. So the latter may be considered as a generalization of the “real part” of Thurston’s construction.

Alexandrov-Fenchel inequalities for convex hypersurfaces with free boundary in a ball

In this paper we first introduce quermassintegrals for free boundary hypersurfaces in the $(n+1)$-dimensional Euclidean unit ball. Then we solve some related isoperimetric type problems for convex free boundary hypersurfaces, which lead to new Alexandrov-Fenchel inequalities. In particular, for $n=2$ we obtain a Minkowski-type inequality and for $n=3$ we obtain an optimal Willmore-type inequality. To prove these estimates, we employ a specifically designed locally constrained inverse harmonic mean curvature flow with free boundary.

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.

A fully nonlinear locally constrained anisotropic curvature flow

Given a smooth positive function F∈C∞(Sn) such that the square of its positive 1-homogeneous extension on Rn+1∖{0} is uniformly convex, the Wulff shape WF is a smooth uniformly convex body in the Euclidean space Rn+1 with F being the support function of the boundary ∂WF. In this paper, we introduce the fully nonlinear locally constrained anisotropic curvature flow ∂∂tX=(1−Ek1/kσF)νF,k=2,…,nin the Euclidean space, where Ek denotes the normalized kth anisotropic mean curvature with respect to the Wulff shape WF, σF the anisotropic support function and νF the outward anisotropic unit normal of the evolving hypersurface. We show that starting from a smooth, closed and strictly convex hypersurface in Rn+1 (n≥2), the smooth solution of the flow exists for all positive time and converges smoothly and exponentially to a scaled Wulff shape. A nice feature of this flow is that it improves a certain isoperimetric ratio. Therefore by the smooth convergence of the above flow, we provide a new proof of a class of the Alexandrov–Fenchel inequalities for anisotropic mixed volumes of smooth convex domains in the Euclidean space.

Reverse Alexandrov–Fenchel inequalities for zonoids

The Alexandrov–Fenchel inequality bounds from below the square of the mixed volume [Formula: see text] of convex bodies [Formula: see text] in [Formula: see text] by the product of the mixed volumes [Formula: see text] and [Formula: see text]. As a consequence, for integers [Formula: see text] with [Formula: see text] the product [Formula: see text] of suitable powers of the volumes [Formula: see text] of the convex bodies [Formula: see text], [Formula: see text], is a lower bound for the mixed volume [Formula: see text], where [Formula: see text] is the multiplicity with which [Formula: see text] appears in the mixed volume. It has been conjectured by Betke and Weil that there is a reverse inequality, that is, a sharp upper bound for the mixed volume [Formula: see text] in terms of the product of the intrinsic volumes [Formula: see text], for [Formula: see text]. The case where [Formula: see text], [Formula: see text], [Formula: see text] has recently been settled by the present authors (2020). The case where [Formula: see text], [Formula: see text], [Formula: see text] has been treated by Artstein-Avidan et al. under the assumption that [Formula: see text] is a zonoid and [Formula: see text] is the Euclidean unit ball. The case where [Formula: see text], [Formula: see text] is the unit ball and [Formula: see text] are zonoids has been considered by Hug and Schneider. Here, we substantially generalize these previous contributions, in cases where most of the bodies are zonoids, and thus we provide further evidence supporting the conjectured reverse Alexandrov–Fenchel inequality. The equality cases in all considered inequalities are characterized. More generally, stronger stability results are established as well.

Weighted Alexandrov–Fenchel inequalities in hyperbolic space and a conjecture of Ge, Wang, and Wu

We consider a conjecture made by Ge, Wang, and Wu regarding weighted Alexandrov–Fenchel inequalities for horospherically convex hypersurfaces in hyperbolic space (a bound, for some physically motivated weight function, of the weighted integral of the k k th mean curvature in terms of the area of the hypersurface). We prove an inequality very similar to the conjectured one. Moreover, when k k is zero and the ambient space has dimension three, we give a counterexample to the conjectured 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.

Higher Lelong Numbers and Convex Geometry

We prove the reversed Alexandrov–Fenchel inequality for mixed Monge–Ampere masses of plurisubharmonic functions, which generalizes a result of Demailly and Pham. As applications to convex geometry, this gives a complex analytic proof of the reversed Alexandrov–Fenchel inequality for mixed covolumes, which generalizes recent results in convex geometry of Kaveh–Khovanskii, Khovanskii–Timorin, Milman–Rotem and Schneider on reversed (or complemented) Brunn–Minkowski and Alexandrov–Fenchel inequalities. Also for toric plurisubharmonic functions in the Cegrell class, we confirm Demailly’s conjecture on the convergence of higher Lelong numbers under the canonical approximation.