Isoperimetric Profile Research Articles

Isoperimetric Control of an Expanding Set

Consider a contaminated region in a 2-dimensional plane that is expanding with unit normal speed. Our goal is to apply a control function (e.g. pesticide) along the boundary of this region to reduce the contaminated area to zero in the shortest possible time, while consistently clearing a fixed amount of land per unit time. It has been proven that when the initial contamination forms a convex set, an optimal eradication strategy requires applying pesticide to the boundary segments with the highest curvature at each moment. This is equivalent to adopting a myopic strategy that minimizes the perimeter of the contaminated region at each time step. This result provided insight for developing numerical methods to compute approximations of these optimal strategies for convex sets. Namely, our simulation consists in alternating between computing the isoperimetric (perimeter-minimizing) profile with a convex constraint and then dilating the resulting set by a radius equal to the time mesh. This process is repeated until the shape reaches a circle, which can then be trivially scaled down to a point. The isoperimetric profile is obtained through morphological opening, an erosion-dilation operator commonly used in image processing for noise removal. A major open problem remains: a full characterization of optimal eradication strategies for initial contaminations of general shapes. We have applied the above numerical scheme with non-convex constraints to gain some intuition in this broader context. This may entail replacing the morphological opening operator with a more general algorithm relying on the medial axis of a shape. It turns out that, in some configurations, it is more effective to split the contamination into several components and shrink them separately.

Building prescribed quantitative orbit equivalence with the integers

Two groups are orbit equivalent if they both admit an action on a same probability space that share the same orbits. In particular, the Ornstein–Weiss theorem implies that all infinite countable amenable groups are orbit equivalent to the group of integers. To refine this notion between infinite countable amenable groups, Delabie, Koivisto, Le Maître and Tessera introduced a quantitative version of orbit equivalence. They furthermore obtained obstructions to the existence of such equivalence using the isoperimetric profile. In this article, we offer to answer the inverse problem (find a group being orbit equivalent to a prescribed group with prescribed quantification) in the case of the group of integers using the so called Følner tiling shifts introduced by Delabie et al. To do so, we use the diagonal products defined by Brieussel and Zheng giving groups with prescribed isoperimetric profile.

Local behavior of the Eden model on graphs and tessellations of manifolds

The Eden Model in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^n$$\\end{document} constructs a blob as follows: initially a single unit hypercube is infected, and each second a hypercube adjacent to the infected ones is selected randomly and infected. Manin, Roldán, and Schweinhart investigated the topology of the Eden model in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^{n}$$\\end{document} by considering the possible shapes which can appear on the boundary. In particular, they give probabilistic lower bounds on the Betti numbers of the Eden model. In this paper, we prove analogous results for the Eden model on any infinite, vertex-transitive, locally finite graph: with high probability as time goes to infinity, every “possible” subgraph (with mild conditions on what “possible” means) occurs on the boundary of the Eden model at least a number of times proportional to an isoperimetric profile of the graph. Using this, we can extend the results about the topology of the Eden model to non-Euclidean spaces, such as hyperbolic n-space and universal covers of certain Riemannian manifolds.

On the isoperimetric profile of the hypercube

On the isoperimetric profile of the hypercube

Invariant constant mean curvature tubes around a horizontal geodesic in [formula omitted]-spaces

We consider constant mean curvature surfaces invariant by a continuous group of isometries and lying at bounded distance from a horizontal geodesic on any homogeneous 3-manifold E(κ,τ) with isometry group of dimension 4. These surfaces are called horizontal tubes. We show that they foliate E(κ,τ) minus one or two horizontal geodesics provided that (1−x02)κ+4τ2≤0, where x0≈0.833557. We also describe precisely how horizontal and vertical geodesics get deformed by Daniel's sister correspondence and conclude that the family of horizontal tubes is preserved by the correspondence. These tubes are topologically tori in S2×R and Berger spheres, in which case we compute their conformal type and analyze numerically their isoperimetric profiles.

Lower bounds for isoperimetric profiles and Yamabe constants

We estimate explicit lower bounds for the isoperimetric profiles of the Riemannian product of a compact manifold and the Euclidean space with the flat metric, (Mm×Rn,g+gE), m,n>1. In particular, we introduce a lower bound for the isoperimetric profile of Mm×Rn for regions of large volume and we improve on previous estimates of lower bounds for the isoperimetric profiles of S2×R2, S3×R2, S2×R3. We also discuss some applications of these results in order to improve known lower bounds for the Yamabe invariant of certain product manifolds.

Asymptotic isoperimetry on non collapsed spaces with lower Ricci bounds

This paper studies sharp and rigid isoperimetric comparison theorems and asymptotic isoperimetric properties for small and large volumes on N-dimensional RCD(K,N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{RCD}(K,N)$$\\end{document} spaces (X,d,HN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(X,\ extsf {d},\\mathscr {H}^N)$$\\end{document}. Moreover, we obtain almost regularity theorems formulated in terms of the isoperimetric profile and enhanced consequences at the level of several functional inequalities. Most of our statements seem to be new even in the classical setting of smooth, non compact manifolds with lower Ricci curvature bounds. The synthetic theory plays a key role via compactness and stability arguments.

Separation profile, isoperimetry, growth and compression

We give lower and upper bounds for the separation profile (introduced by Benjamini, Schramm & Timár) for various graphs using isoperimetric profile, volume growth and Hilbertian compression. For graphs which have polynomial isoperimetry and growth, we show that the separation profile is bounded between n a and n b for some a,b∈(0,1). For many amenable groups, we prove a lower bound of n/log(n) a for some a>1, and for groups admitting “good” embeddings into an ℓ p space we prove an upper bound of n/log(n) b for some b∈(0,1). We show that solvable groups of exponential growth cannot have a separation profile bounded above by a sublinear power function. We also introduce a notion of local separation, with applications for percolation clusters of ℤ d and graphs which have polynomial isoperimetry and growth.

On the existence of isoperimetric regions in manifolds with nonnegative Ricci curvature and Euclidean volume growth

In this paper we provide new existence results for isoperimetric sets of large volume in Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth. We find sufficient conditions for their existence in terms of the geometry at infinity of the manifold. As a byproduct we show that isoperimetric sets of big volume always exist on manifolds with nonnegative sectional curvature and Euclidean volume growth. Our method combines an asymptotic mass decomposition result for minimizing sequences, a sharp isoperimetric inequality on nonsmooth spaces, and the concavity property of the isoperimetric profile. The latter is new in the generality of noncollapsed manifolds with Ricci curvature bounded below.

Isoperimetric inequalities in unbounded convex bodies

We consider the problem of minimizing the relative perimeter under a volume constraint in an unbounded convex body C ⊂ R n C\subset \mathbb {R}^{n} , without assuming any further regularity on the boundary of C 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 C , we prove existence of isoperimetric regions in a generalized sense. By an approximation argument we show the strict concavity of the isoperimetric profile and, consequently, the connectedness of generalized isoperimetric regions. We also focus on the cases of small as well as of large volumes; in particular we show existence of isoperimetric regions with sufficiently large volumes, for special classes of unbounded convex bodies. We finally address some questions about isoperimetric rigidity and analyze the asymptotic behavior of the isoperimetric profile in connection with the notion of isoperimetric dimension.

The isoperimetric problem in 2d domains without necks

We give a complete characterization of all isoperimetric sets contained in a domain of the Euclidean plane, that is bounded by a Jordan curve and satisfies a no neck property. Further, we prove that the isoperimetric profile of such domain is convex above the volume of the largest ball contained in it, and that its square is globally convex.

Some remarks on the Sobolev inequality in Riemannian manifolds

We investigate Sobolev and Hardy inequalities, specifically weighted Minerbe’s type estimates, in noncompact complete connected Riemannian manifolds whose geometry is described by an isoperimetric profile. In particular, we assume that the manifold satisfies the p p -hyperbolicity property, stated in terms of a necessary integral Dini condition on the isoperimetric profile. Our method seems to us to combine sharply the knowledge of the isoperimetric profile and the optimal Bliss type Hardy inequality depending on the geometry of the manifold. We recover the well known best Sobolev constant in the Euclidean case.

Random walks on the discrete affine group

We introduce the discrete affine group of a regular tree as a finitely generated subgroup of the affine group. We describe the Poisson boundary of random walks on it as a space of configurations. We compute isoperimetric profile and Hilbert compression exponent of the group. We also discuss metric relationship with some lamplighter groups and lamplighter graphs.

Isoperimetric profiles and random walks on some groups defined by piecewise actions

We study the isoperimetric and spectral profiles of certain families of finitely generated groups defined via actions on labelled Schreier graphs and simple gluing of such. In one of our simplest constructions—the pocket-extension of a group G—this leads to the study of certain finitely generated subgroups of the full permutation group \({\mathbb {S}}(G\cup \{*\})\). Some sharp estimates are obtained while many challenging questions remain.

Uniform Lipschitz continuity of the isoperimetric profile of compact surfaces under normalized Ricci flow

We show that the isoperimetric profile h g ( t ) ( ξ ) h_{g(t)}(\xi ) of a compact Riemannian manifold ( M , g ) (M,g) is jointly continuous when metrics g ( t ) g(t) vary continuously. We also show that, when M M is a compact surface and g ( t ) g(t) evolves under normalized Ricci flow, h g ( t ) 2 ( ξ ) h^2_{g(t)}(\xi ) is uniform Lipschitz continuous and hence h g ( t ) ( ξ ) h_{g(t)}(\xi ) is uniform locally Lipschitz continuous.

Speed of random walks, isoperimetry and compression of finitely generated groups

We give a solution to the inverse problem (given a prescribed function, find a corresponding group) for large classes of speed, entropy, isoperimetric profile, return probability and $L_p$-compression functions of finitely generated groups. For smaller classes, we give solutions among solvable groups of exponential volume growth. As corollaries, we prove a recent conjecture of Amir on joint evaluation of speed and entropy exponents and we obtain a new proof of the existence of uncountably many pairwise non-quasi-isometric solvable groups, originally due to Cornulier and Tessera. We also obtain a formula relating the $L_p$-compression exponent of a group and its wreath product with the cyclic group for$p$ in $[1,2]$.

On the Hamilton's isoperimetric ratio in complete Riemannian manifolds of finite volume

We study a family of geometric variational functionals introduced by Hamilton, and considered later by Daskalopulos, Sesum, Del Pino and Hsu, in order to understand the behavior of maximal solutions of the Ricci flow both in compact and noncompact complete Riemannian manifolds of finite volume. The case of dimension two has some peculiarities, which force us to use different ideas from the corresponding higher-dimensional case. Under some natural restrictions, we investigate sufficient and necessary conditions which allow us to show the existence of connected regions with a connected complementary set (the so-called “separating regions”). In dimension higher than two, the associated problem of minimization is reduced to an auxiliary problem for the isoperimetric profile (with the corresponding investigation of the minimizers). This is possible via an argument of compactness in geometric measure theory valid for the case of complete finite volume manifolds. Moreover, we show that the minimum of the separating variational problem is achieved by an isoperimetric region. The dimension two requires different techniques of proof. The present results develop a definitive theory, which allows us to circumvent the shortening curve flow approach of the above mentioned authors at the cost of some applications of the geometric measure theory and of the Ascoli-Arzela's Theorem.

Differentiability properties of the isoperimetric profile in complete noncompact Riemannian manifolds with $$C^0$$-locally asymptotic bounded geometry

For a complete noncompact connected Riemannian manifold with bounded geometry $$M^n$$ , we prove that the isoperimetric profile function $$I_{M^n}$$ is twice differentiable almost everywhere. Moreover, we show that a differential inequality is satisfied by $$I_M$$ ; extending in this way well-known results for compact manifolds due to Bavard and Pansu (Ann Sci Ecole Norm Sup :479–490, 1986 ), to this class of noncompact complete Riemannian manifolds with bounded geometry. Here for $$C^0$$ -locally asymptotic bounded geometry we mean that for all pointed sequences $$p_j\in M$$ diverging at infinity the sequence of pointed Riemannian manifolds $$(M,p_j,g)$$ sub-converge in $$C^0$$ topology to a limit manifold $$(M_\infty , g_\infty , p_\infty )$$ that we assume to be at least of class $$C^2$$ .

Medial Axis Isoperimetric Profiles

AbstractRecently proposed as a stable means of evaluating geometric compactness, the isoperimetric profile of a planar domain measures the minimum perimeter needed to inscribe a shape with prescribed area varying from 0 to the area of the domain. While this profile has proven valuable for evaluating properties of geographic partitions, existing algorithms for its computation rely on aggressive approximations and are still computationally expensive. In this paper, we propose a practical means of approximating the isoperimetric profile and show that for domains satisfying a “thick neck” condition, our approximation is exact. For more general domains, we show that our bound is still exact within a conservative regime and is otherwise an upper bound. Our method is based on a traversal of the medial axis which produces efficient and robust results. We compare our technique with the state‐of‐the‐art approximation to the isoperimetric profile on a variety of domains and show significantly tighter bounds than were previously achievable.

Symmetrization inequalities for probability metric spaces with convex isoperimetric profile

We obtain symmetrization inequalities on probability metric spaces with convex isoperimetric profile which incorporate in their formulation the isoperimetric estimator and that can be applied to provide a unified treatment of sharp Sobolev-Poincare and Nash inequalities.