Isoperimetric Research Articles

Discrete isoperimetric problems in spaces of constant curvature

AbstractThe aim of this paper is to prove isoperimetric inequalities for simplices and polytopes with vertices in Euclidean, spherical and hyperbolic d‐space. In particular, we find the minimal volume d‐dimensional hyperbolic simplices and spherical tetrahedra of a given inradius. Furthermore, we investigate the properties of maximal volume spherical and hyperbolic polytopes with vertices with a given circumradius, and the hyperbolic polytopes with vertices with a given inradius and having a minimal volume or minimal total edge length. Finally, for any , we investigate the properties of Euclidean simplices and polytopes with vertices having a fixed inradius and a minimal volume of its k‐skeleton. The main tool of our investigation is Euclidean, spherical and hyperbolic Steiner symmetrization.

Lp$L_p$‐cosine transform of the (p,q)${(p,q)}$‐th dual curvature measure

AbstractThe ‐th dual curvature measure was introduced by Lutwak, Yang and Zhang. In this paper, we study the cosine transform of the ‐th dual curvature measure which defines a new convex body. We prove that the new convex body unifies the Petty projection body and the centroid body. The corresponding affine isoperimetric inequality is also obtained. It is an extension of the known Petty projection inequality.

An isoperimetric inequality for the perturbed Robin bi-Laplacian in a planar exterior domain

We introduce the perturbed two-dimensional Robin bi-Laplacian in the exterior of a planar bounded simply-connected C2-smooth open set. The considered lower-order perturbation corresponds to tension. We prove that the essential spectrum of this operator coincides with the positive semi-axis and that the negative discrete spectrum is non-empty if the boundary parameter is negative. As the main result, we obtain an isoperimetric inequality for the lowest eigenvalue of such a perturbed Robin bi-Laplacian with a negative boundary parameter in the exterior of a bounded convex planar set under the constraint on the maximum of the curvature of the boundary with the maximizer being the exterior of the disk. The isoperimetric inequality is proved under the assumption that to the lowest eigenvalue for the exterior of the disk corresponds a radial eigenfunction. We provide a sufficient condition in terms of the tension parameter and the radius for this property to hold.

A quantitative Bucur–Henrot inequality

AbstractIn this paper, we prove a quantitative version of the isoperimetric inequality involving the second non‐trivial eigenvalue of the Laplacian with Neumann boundary condition established by Bucur and Henrot [5].

Crux and Long Cycles in Graphs

We introduce a notion of the crux of a graph $G$, measuring the order of a smallest dense subgraph in $G$. This simple-looking notion leads to some generalizations of known results about cycles, offering an interesting paradigm of “replacing average degree by crux.” In particular, we prove that every graph contains a cycle of length linear in its crux. Long proved that every subgraph of a hypercube $Q^m$ (resp., discrete torus $C_3^m$) with average degree $d$ contains a path of length $2^{d/2}$ (resp., $2^{d/4}$) and conjectured that there should be a path of length $2^{d}-1$ (resp., $3^{d/2}-1$). As a corollary of our result, together with isoperimetric inequalities, we close these exponential gaps giving asymptotically optimal bounds on long paths in hypercubes, discrete tori, and more generally Hamming graphs. We also consider random subgraphs of $C_4$-free graphs and hypercubes, proving near optimal lower bounds on the lengths of long cycles.

The isoperimetric problem on Riemannian manifolds via Gromov–Hausdorff asymptotic analysis

In this paper, we prove the existence of isoperimetric regions of any volume in Riemannian manifolds with Ricci bounded below assuming Gromov–Hausdorff asymptoticity to the suitable simply connected model of constant sectional curvature. The previous result is a consequence of a general structure theorem for perimeter-minimizing sequences of sets of fixed volume on noncollapsed Riemannian manifolds with a lower bound on the Ricci curvature. We show that, without assuming any further hypotheses on the asymptotic geometry, all the mass and the perimeter lost at infinity, if any, are recovered by at most countably many isoperimetric regions sitting in some (possibly nonsmooth) Gromov–Hausdorff limits at infinity. The Gromov–Hausdorff asymptotic analysis allows us to recover and extend different previous existence theorems. While studying the isoperimetric problem in the smooth setting, the nonsmooth geometry naturally emerges, and thus our treatment combines techniques from both the theories.

The Saint‐Venant type isoperimetric inequalities for assessing saturated water storage in lacunary shallow perched aquifers

AbstractThe Dupuit‐Forchheimer (DF) approximation for unconfined groundwater flows is reduced to 2D Poisson's equation (PE), the right hand side of which involves an intensive evapotranspiration (ET) rate. A shallow water table dips inward from a constant piezometric head boundary (closed curve) such that a “dry gap,” demarcated by another closed curve (unknown front), may emerge. Apriori estimates of the volume of the saturated zone and of the “dry gap” area are important for water resources management in drylands. We use the results from the theory of linear elasticity, torsion of elastic bars, for which PE is solved for the Prandtl function. Using the Poincaré metrics with the Gaussian constant curvature , isoperimetric inequalities are obtained for steady‐state DF flows. Conformal moments and confromal radii of the domains and 2‐D Hardy's type inequalities in domains, modeling the Saint Venant bar torsion problem, are involved in the obtained estimates. The studied boundary value problems (BVPs) are nonlinear for ET rates depending on the depth of the water table. In the DF model, the vadose zone (VZ) is considered as a “distributed sink” (similar to a standard “distributed source,” which models recharge to the water table in humid climates). The analytical VZ is collated with one obtained from a numerical solution to BVP for Richards’ equation in a 3‐D saturated‐unsaturated flow. BVPs for Richards’ equation in cylindrical domains, solved by HYDRUS software, give the pressure head, moisture content, Darcian velocity fields, and streamlines.

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.

The Isoperimetric Problem for Regular and Crystalline Norms in $${\mathbb {H}}^1$$

We study the isoperimetric problem for anisotropic perimeter measures on \(\mathbb {R}^3\), endowed with the Heisenberg group structure. The perimeter is associated with a left-invariant norm \(\phi \) on the horizontal distribution. In the case where \(\phi \) is the standard norm in the plane, such isoperimetric problem is the subject of Pansu’s conjecture, which is still unsolved. Assuming some regularity on \(\phi \) and on its dual norm \(\phi ^*\), we characterize \(\mathrm {C}^2\)-smooth isoperimetric sets as the sub-Finsler analogue of Pansu’s bubbles. The argument is based on a fine study of the characteristic set of \(\phi \)-isoperimetric sets and on establishing a foliation property by sub-Finsler geodesics. When \(\phi \) is a crystalline norm, we show the existence of a partial foliation for constant \(\phi \)-curvature surfaces by sub-Finsler geodesics. By an approximation procedure, we finally prove a conditional minimality property for the candidate solutions in the general case (including the case where \(\phi \) is crystalline).

Isoperimetric problem for the first curl eigenvalue

We consider an isoperimetric problem involving the smallest positive and largest negative curl eigenvalues on abstract ambient manifolds, with a focus on the standard model spaces. We in particular show that the corresponding eigenvalues on optimal domains, assuming optimal domains exist, must be simple in the Euclidean and hyperbolic setting. This generalises a recent result by Enciso and Peralta-Salas who showed the simplicity for axisymmetric optimal domains with connected boundary in Euclidean space. We then generalise another recent result by Enciso and Peralta-Salas, namely that the points of any rotationally symmetric optimal domain with connected boundary in Euclidean space which are closest to the symmetry axis must disconnect the boundary, to the hyperbolic setting, as well as strengthen it in the Euclidean case by removing the connected boundary assumption.

Ricci curvature, Bruhat graphs and Coxeter groups

We consider the discrete Ricci curvature for graphs as defined by Schmuckenschläger [Convex geometric analysis, MSRI Publications, 1998] and compute its value for Bruhat graphs associated to finite Coxeter groups. To do so we work with the geometric realization of a finite Coxeter group and a classical result obtained by Dyer in [Compositio Math. 78 (1991), pp. 185–191]. As an application we obtain a bound for the spectral gap of the Bruhat graph of any finite Coxeter group and an isoperimetric inequality for them. Our proofs are case-free.

An inequality for the normal derivative of the Lane–Emden ground state

Abstract We consider Lane–Emden ground states with polytropic index 0 ≤ q - 1 ≤ 1 {0\leq q-1\leq 1} , that is, minimizers of the Dirichlet integral among L q {L^{q}} -normalized functions. Our main result is a sharp lower bound on the L 2 {L^{2}} -norm of the normal derivative in terms of the energy, which implies a corresponding isoperimetric inequality. Our bound holds for arbitrary bounded open Lipschitz sets Ω ⊂ ℝ d {\Omega\subset\mathbb{R}^{d}} , without assuming convexity.

Continuum limits of discrete isoperimetric problems and Wulff shapes in lattices and quasicrystal tilings

We prove discrete-to-continuum convergence of interaction energies defined on lattices in the Euclidean space (with interactions beyond nearest neighbours) to a crystalline perimeter, and we discuss the possible Wulff shapes obtainable in this way. Exploiting the “multigrid construction” of quasiperiodic tilings (which is an extension of De Bruijn’s “pentagrid” construction of Penrose tilings) we adapt the same techniques to also find the macroscopical homogenized perimeter when we microscopically rescale a given quasiperiodic tiling.

Linear isoperimetric inequalities for free boundary submanifolds in a geodesic ball

We derive linear isoperimetric inequalities for free boundary submanifolds in a geodesic ball of a Riemannian manifold in terms of the modified volume. It is known that the twice of the area of a free boundary minimal surface in a Euclidean unit ball is equal to the length of its boundary. This can be extended to space forms by using our linear isoperimetric inequalities for the modified volume. Moreover, we obtain a sharp lower bound for the modified volume of free boundary minimal surfaces in a geodesic ball of the upper hemisphere mathbb {S}_+^3. Finally, it is proved that the monotonicity property still holds for the modified volume of any submanifold in a complete simply connected Riemannian manifold with sectional curvature bounded above by a constant.

Sharp Morrey–Sobolev inequalities and eigenvalue problems on Riemannian–Finsler manifolds with nonnegative Ricci curvature

Combining the sharp isoperimetric inequality established by Z. Balogh and A. Kristály [Math. Ann., in press, doi:10.1007/s00208-022-02380-1] with an anisotropic symmetrization argument, we establish sharp Morrey–Sobolev inequalities on [Formula: see text]-dimensional Finsler manifolds having nonnegative [Formula: see text]-Ricci curvature. A byproduct of this method is a Hardy–Sobolev-type inequality in the same geometric setting. As applications, by using variational arguments, we guarantee the existence/multiplicity of solutions for certain eigenvalue problems and elliptic PDEs involving the Finsler–Laplace operator. Our results are also new in the Riemannian setting.

Spectral isoperimetric inequalities for Robin Laplacians on 2-manifolds and unbounded cones

We consider the problem of geometric optimization of the lowest eigenvalue for the Laplacian on a compact, simply-connected two-dimensional manifold with boundary subject to an attractive Robin boundary condition. We prove that in the sub-class of manifolds with the Gauss curvature bounded from above by a constant K_\circ \ge 0 and under the constraint of fixed perimeter, the geodesic disk of constant curvature K_\circ maximizes the lowest Robin eigenvalue. In the same geometric setting, it is proved that the spectral isoperimetric inequality holds for the lowest eigenvalue of the Dirichlet-to-Neumann operator. Finally, we adapt our methods to Robin Laplacians acting on unbounded three-dimensional cones to show that, under a constraint of fixed perimeter of the cross-section, the lowest Robin eigenvalue is maximized by the circular cone.

A sharp isoperimetric inequality for the second eigenvalue of the Robin plate

Among all C^\infty bounded domains with equal volume, we show that the second eigenvalue of the Robin plate is uniquely maximized by an open ball, so long as the Robin parameter lies within a particular range of negative values. Our methodology combines recent techniques introduced by Freitas and Laugesen to study the second eigenvalue of the Robin membrane problem and techniques employed by Chasman to study the free plate problem. In particular, we choose eigenfunctions of the ball as trial functions in the Rayleigh quotient for a general domain; such eigenfunctions are comprised of ultraspherical Bessel andmodified Bessel functions. Much of our work hinges on developing an understanding of delicate properties of these special functions, which may be of independent interest.

Quantitative estimates for the Bakry–Ledoux isoperimetric inequality II

Quantitative estimates for the Bakry–Ledoux isoperimetric inequality II

Isoperimetric Inequalities for the Magnetic Neumann and Steklov Problems with Aharonov–Bohm Magnetic Potential

We discuss isoperimetric inequalities for the magnetic Laplacian on bounded domains of {mathbb {R}}^2 endowed with an Aharonov–Bohm potential. When the flux of the potential around the pole is not an integer, the lowest eigenvalue for the Neumann and the Steklov problems is positive. We establish isoperimetric inequalities for the lowest eigenvalue in the spirit of the classical inequalities of Szegö–Weinberger, Brock and Weinstock, the model domain being a disk with the pole at its center. We consider more generally domains in the plane endowed with a rotationally invariant metric, which include the spherical and the hyperbolic case.

Isoperimetric inequalities in the Brownian plane

We consider the model of the Brownian plane, which is a pointed noncompact random metric space with the topology of the complex plane. The Brownian plane can be obtained as the scaling limit in distribution of the uniform infinite planar triangulation or the uniform infinite planar quadrangulation and is conjectured to be the universal scaling limit of many others random planar lattices. We establish sharp bounds on the probability of having a short cycle separating the ball of radius r centered at the distinguished point from infinity. Then we prove a strong version of the spatial Markov property of the Brownian plane. Combining our study of short cycles with this strong spatial Markov property we obtain sharp isoperimetric bounds for the Brownian plane.