We characterize convex isoperimetric sets in the Heisenberg group endowed with horizontal perimeter. We first prove Sobolev regularity for a certain class of vector fields in the plane with bounded variation, related to the curvature equations. Then, by an approximation-reparameterization argument, we show that the boundary of convex isoperimetric sets is foliated by geodesics of the Carnot-Caratheodory distance.
In this paper we prove that isoperimetric sets in three-dimensional homogeneous spaces diffeomorphic to $\mathbb{R}^3$ are topological balls. We also prove that in three-dimensional homogeneous spheres isopermetric sets are either two-spheres or symmetric genus-one tori. We then apply our first result to the three-dimensional Heisenberg group $\mathsf{H}^1$, characterizing the isoperimetric sets and constants for a family of Riemannian adapted metrics. Using $Γ$-convergence of the perimeter functionals, we also settle an isoperimetric conjecture in $\mathsf{H}^1$ posed by P.Pansu.
In the context of sub-Riemannian Heisenberg groups Hn, n \geq 1, we shall study Isoperimetric Profiles, which are closed compact hypersurfaces having constant horizontal mean curvature, very similar to ellipsoids. Our main goal is to study the stability of Isoperimetric Profiles.
We prove that geodesic balls centered at some base point are isoperimetric in the real hyperbolic space $H_{\mathbb R}^n$ endowed with a smooth, radial, strictly log-convex density on the volume and perimeter. This is an analogue of the result by G. R. Chambers for log-convex densities on $\mathbb R^n$. As an application we prove that in any rank one symmetric space of non-compact type, geodesic balls are isoperimetric in a class of sets enjoying a suitable notion of radial symmetry.
After introducing the sub-Riemannian geometry of the Heisenberg group Hn, n \geq 1, we recall some basics about hypersurfaces endowed with the H-perimeter measure and horizontal Green's formulas. Then, we describe a class of compact closed hypersurfaces of constant horizontal mean curvature called "Isoperimetric Profiles"(they are not CC-balls!); see Section 2.1. Our main purpose is to study a closed eigenvalue problem on Isoperimetric Profiles, i.e. LHS ϕ+ λϕ= 0, where LHS is a 2nd order horizontal tangential operator analogous to the Laplace-Beltrami operator; see Section 1.5. This is done s…
We present a new and direct proof of the local Neumann isoperimetric inequality on convex domains of a Riemannian manifold with Ricci curvature bounded below.
Let $D$ be a closed disk centered at the origin in the horizontal hyperplane $\{t=0\}$ of the sub-Riemannian Heisenberg group $\hh^n$, and $C$ the vertical cylinder over $D$. We prove that any finite perimeter set $E$ such that $D\subset E\subset C$ has perimeter larger than or equal to the one of the rotationally symmetric sphere with constant mean curvature of the same volume, and that equality holds only for the spheres using a recent result by Monti and Vittone [12].
In this paper we consider the problem of minimizing the relative perimeter under a volume constraint in the interior of a conically bounded convex set, i.e., an unbounded convex body admitting an \emph{exterior} asymptotic cone. Results concerning existence of isoperimetric regions, the behavior of the isoperimetric profile for large volumes, and a characterization of isoperimetric regions of large volume in conically bounded convex sets of revolution is obtained.
In the first part of the paper we survey some nonlocal flows of convex plane curves ever studied so far and discuss properties of the flows related to enclosed area and length, especially the isoperimetric ratio and the isoperimetric difference. We also study a new nonlocal flow of convex plane curves and discuss its evolution behavior. In the second part of the paper we discuss necessary and sufficient conditions (in terms of the (mixed) isoperimetric ratio or (mixed) isoperimetric difference) for two convex closed curves to be homothetic or parallel.
The convex body isoperimetric conjecture in the plane asserts that the least perimeter to enclose given area inside a unit disk is greater than inside any other convex set of area $π$. In this note we confirm two cases of the conjecture: domains symmetric to both coordinate axes and perturbations of unit disk.
In this paper we consider the isoperimetric profile of convex cylinders $K\times\mathbb{R}^q$, where $K$ is an $m$-dimensional convex body, and of cylindrically bounded convex sets, i.e, those with a relatively compact orthogonal projection over some hyperplane of $\mathbb{R}^{n+1}$, asymptotic to a right convex cylinder of the form $K\times\mathbb{R}$, with $K\subset\mathbb{R}^n$. Results concerning the concavity of the isoperimetric profile, existence of isoperimetric regions, and geometric descriptions of isoperimetric regions for small and large volumes are obtained.
In this paper we consider the problem of minimizing the relative perimeter under a volume constraint in the interior of a convex body, i.e., a compact convex set in Euclidean space with interior points. We shall not impose any regularity assumption on the boundary of the convex set. Amongst other results, we shall prove the equivalence between Hausdorff and Lipschitz convergence, the continuity of the isoperimetric profile with respect to the Hausdorff distance,and the convergence in Hausdorff distance of sequences of isoperimetric regions and their free boundaries. We shall also describe the …
We consider the problem of minimizing the relative perimeter under a volume constraint in an unbounded convex body $C\subset \mathbb{R}^{n+1}$, without assuming any further regularity on the boundary of $C$. Motivated by an example of an unbounded convex body with null isoperimetric profile, we introduce the concept of unbounded convex body with uniform geometry. We then provide a handy characterization of the uniform geometry property and, by exploiting the notion of asymptotic cylinder of $C$, we prove existence of isoperimetric regions in a generalized sense. By an approximation argument we…
We give a sharp lower bound on the area of a domain that can be enclosed by a closed embedded $λ$-convex curve of a given length on the Lobachevsky plane.
We reinterpret the renormalized volume as the asymptotic difference of the isoperimetric profiles for convex co-compact hyperbolic 3-manifolds. By similar techniques we also prove a sharp Minkowski inequality for horospherically convex sets in $\mathbb{H}^3$. Finally, we include the classification of stable constant mean curvature surfaces in regions bounded by two geodesic planes in $\mathbb{H}^3$ or in cyclic quotients of $\mathbb{H}^3$.
The purpose of this paper is to analyze the isoperimetric inequality for symmetric log-convex probability measures on the line. Using geometric arguments we first re-prove that extremal sets in the isoperimetric inequality are intervals or complement of intervals (a result due to Bobkov and Houdré). Then we give a quantitative form of the isoperimetric inequality, leading to a somehow anomalous behavior. Indeed, it could be that a set is very close to be optimal, in the sense that the isoperimetric inequality is almost an equality, but at the same time is very far (in the sense of the symmetri…
We prove that the isoperimetric profile of a convex domain $Ω$ with compact closure in a Riemannian manifold $(M^{n+1},g)$ satisfies a second order differential inequality which only depends on the dimension of the manifold and on a lower bound on the Ricci curvature of $Ω$. Regularity properties of the profile and topological consequences on isoperimetric regions arise naturally from this differential point of view. Moreover, by integrating the differential inequality we obtain sharp comparison theorems: not only can we derive an inequality which should be compared with Lévy-Gromov Inequality…
We extend several Cheeger-type isoperimetric bounds for convex sets in Euclidean space, due to Bobkov and Kannan-Lovász-Simonovits, to Riemannian manifolds having non-negative Ricci curvature. In order to extend Bobkov's bound, we require in addition an upper bound on the sectional curvature of the space, which permits us to use comparison tools in Cartan-Alexandrov-Toponogov (or CAT) spaces. Along the way, we also quantitatively improve our previous result that weak concentration assumptions imply a Cheeger-type isoperimetric bound, to a sharp bound with respect to all parameters.
In this paper we solve several reverse isoperimetric problems in the class of $λ$-convex bodies, i.e., convex bodies whose curvature at each point of their boundary is bounded below by some $λ> 0$. We give an affirmative answer in $\mathbb{R}^3$ to a conjecture due to Borisenko which states that the $λ$-convex lens, i.e., the intersection of two balls of radius $1/λ$, is the unique minimizer of volume among all $λ$-convex bodies of given surface area. Also, we prove a reverse inradius inequality: in model spaces of constant curvature and arbitrary dimension, we show that the $λ$-convex lens (p…
We completely characterize isoperimetric regions in R^n with density e^h, where h is convex, smooth, and radially symmetric. In particular, balls around the origin constitute isoperimetric regions of any given volume, proving the Log-Convex Density Conjecture due to Kenneth Brakke.
Given a positive lower semi-continuous density $f$ on $\mathbb{R}^2$ the weighted volume $V_f:=f\mathscr{L}^2$ is defined on the $\mathscr{L}^2$-measurable sets in $\mathbb{R}^2$. The $f$-weighted perimeter of a set of finite perimeter $E$ in $\mathbb{R}^2$ is written $P_f(E)$. We study minimisers for the weighted isoperimetric problem \[ I_f(v):=\inf\Big\{ P_f(E):E\text{ is a set of finite perimeter in }\mathbb{R}^2\text{ and }V_f(E)=v\Big\} \] for $v>0$. Suppose $f$ takes the form $f:\mathbb{R}^2\rightarrow(0,+\infty);x\mapsto e^{h(|x|)}$ where $h:[0,+\infty)\rightarrow\mathbb{R}$ is a non-d…
In this paper, we deals with isoperimetric-type inequalities for closed convex curves in the Euclidean plane R^2. We derive a family of parametric inequalities involving the following geometric functionals associated to a given convex curve with a simple Fourier series proof: length, area of the region included by the curve, area of the domain enclosed by the locus of curvature centers and integral of the radius of curvature. By using our isoperimetric-type inequalities, we also obtain some new geometric Bonnesen-type inequalities. Furthermore we investigate stability properties of such inequa…
In this paper, we study contact forms on the three- dimensional Heisenberg manifold with its standard CR structure. We discover that the $Q'$-curvature, introduced by Branson, Fontana and Morpurgo [BFM13] on the CR three-sphere and then generalized to any pseudo-Einstein CR three manifold by Case and Yang [CY95], controls the isoperimetric inequality on such a CR-manifold. To show this, we first prove that the nonnegative Webster curvature at infinity deduces that the metric is normal, which is analogous to the behavior on a Riemannian four-manifold.
In this paper, we prove a class of weighted isoperimetric inequalities for bounded domains in hyperbolic space by using the isoperimetric inequality with log-convex density in Euclidean space. As a consequence, we remove the horo-convex assumption of domains in a weighted isoperimetric inequality proved by Scheuer-Xia. Furthermore, we prove weighted isoperimetric inequalities for star-shaped domains in warped product manifolds. Particularly, we obtain a weighted isoperimetric inequality for star-shaped hypersurfaces lying outside a certain radial coordinate slice in the anti-de Sitter-Schwarzs…
In this paper we revisit the anisotropic isoperimetric and the Brunn-Minkowski inequalities for convex sets. The best known constant $C(n)=Cn^{7}$ depending on the space dimension $n$ in both inequalities is due to Segal [\ref{bib:Seg.}]. We improve that constant to $Cn^6$ for convex sets and to $Cn^5$ for centrally symmetric convex sets. We also conjecture, that the best constant in both inequalities must be of the form $Cn^2,$ i.e., quadratic in $n.$ The tools are the Brenier's mapping from the theory of mass transportation combined with new sharp geometric-arithmetic mean and some algebraic…
In this paper, we establish some sharp inequalities between the volume and the integral of the $k$-th mean curvature for $k+1$-convex domains in the Euclidean space. The results generalize the classical Alexandrov-Fenchel inequalities for convex domains. Our proof utilizes the method of optimal transportation.
We prove some old and new isoperimetric inequalities with the best constant using the ABP method applied to an appropriate linear Neumann problem. More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (also called densities) in open convex cones of $\mathbb{R}^n$. Our result applies to all nonnegative homogeneous weights satisfying a concavity condition in the cone. Remarkably, Euclidean balls centered at the origin (intersected with the cone) minimize the weighted isoperimetric quotient, even if all our weights are nonradial ---except for the constant ones. W…
We study a family of spheres with constant mean curvature (CMC) in the Riemannian Heisenberg group $H^1$. These spheres are conjectured to be the isoperimetric sets of $H^1$. We prove several results supporting this conjecture. We also focus our attention on the sub-Riemannian limit.
Employing the affine normal flow, we prove a stability version of the $p$-affine isoperimetric inequality for $p\geq1$ in $\mathbb{R}^2$ in the class of origin-symmetric convex bodies. That is, if $K$ is an origin-symmetric convex body in $\mathbb{R}^2$ such that it has area $π$ and its $p$-affine perimeter is close enough to the one of an ellipse with the same area, then, after applying a special linear transformation, $K$ is close to an ellipse in the Hausdorff distance.
We consider the class of $λ$-concave bodies in $\mathbb R^{n+1}$; that is, convex bodies with the property that each of their boundary points supports a tangent ball of radius $1/λ$ that lies locally (around the boundary point) inside the body. In this class we solve a reverse isoperimetric problem: we show that the convex hull of two balls of radius $1/λ$ (a sausage body) is a unique volume minimizer among all $λ$-concave bodies of given surface area. This is in a surprising contrast to the standard isoperimetric problem for which, as it is well-known, the unique maximizer is a ball. We solve…
For compact Riemannian manifolds with convex boundary, B.White proved the following alternative: Either there is an isoperimetric inequality for minimal hypersurfaces or there exists a closed minimal hypersurface, possibly with a small singular set. There is the natural question if a similar result is true for submanifolds of higher codimension. Specifically, B.White asked if the non-existence of an isoperimetric inequality for k-varifolds implies the existence of a nonzero, stationary, integral k-varifold. We present examples showing that this is not true in codimension greater than two. The …
Let $Ω$ be a measurable Euclidean set in $\mathbb{R}^{n}$ that is symmetric, i.e. $Ω=-Ω$, such that $Ω\times\mathbb{R}$ has the smallest Gaussian surface area among all measurable symmetric sets of fixed Gaussian volume. We conclude that either $Ω$ or $Ω^{c}$ is convex. Moreover, except for the case $H(x)=\langle x,N(x)\rangle+λ$ with $H\geq0$ and $λ<0$, we show there exist a radius $r>0$ and an integer $0\leq k\leq n-1$ such that after applying a rotation, the boundary of $Ω$ must satisfy $\partialΩ= rS^{k}\times\mathbb{R}^{n-k-1}$, with $\sqrt{n-1}\leq r\leq\sqrt{n+1}$ when $k\geq1$. Here $S…
We study the generalization of the Willmore functional for surfaces in the three-Heisenberg group. Its construction is based on the spectral theory of the Dirac operator coming to the Weierstrass representation of surfaces (see math.DG/0503707). By using surfaces of revolution we demonstrate that it resembles the Willmore functional for surfaces in the Euclidean space in many geometrical respects. We also observe the relation of these functionals to the isoperimetric problem.
We review and extend here some recent results on the existence of minimal surfaces and isoperimetric sets in non homogeneous and anisotropic periodic media. We also describe the qualitative properties of the homogenized surface tension, also known as stable norm (or minimal action) in Weak KAM theory. In particular we investigate its strict convexity and differentiability properties.
We consider a nilpotent Lie group with a bracket-generating distribution $\hhh$ and an asymmetric left-invariant norm $\|\cdot\|_K$ induced by a convex body $K\subseteq\hhh$ containing $0$ in its interior. In this paper we prove the existence of minimizers of the perimeter functional $P_K$ associated to $\|\cdot\|_K$ under a volume (Haar measure) constraint. Our result generalizes the one of Leonardi and Rigot for Carnot groups.
The Orlicz-Brunn-Minkowski theory receives considerable attention recently, and many results in the $L_p$-Brunn-Minkowski theory have been extended to their Orlicz counterparts. The aim of this paper is to develop Orlicz $L_ϕ$ affine and geominimal surface areas for single convex body as well as for multiple convex bodies, which generalize the $L_p$ (mixed) affine and geominimal surface areas -- fundamental concepts in the $L_p$-Brunn-Minkowski theory. Our extensions are different from the general affine surface areas by Ludwig (in Adv. Math. 224 (2010)). Moreover, our definitions for Orlicz $…
We study the isoperimetric problem for anisotropic left-invariant perimeter measures on $\mathbb R^3$, endowed with the Heisenberg group structure. The perimeter is associated with a left-invariant norm $ϕ$ on the horizontal distribution. We first prove a representation formula for the $ϕ$-perimeter of regular sets and, assuming some regularity on $ϕ$ and on its dual norm $ϕ^*$, we deduce a foliation property by sub-Finsler geodesics of $\mathrm C^2$-smooth surfaces with constant $ϕ$-curvature. We then prove that the characteristic set of $\mathrm C^2$-smooth surfaces that are locally extremal…
