It is shown that if $C$ is an $n$-dimensional convex body then there is an affine image $\widetilde C$ of $C$ for which
$${|\partial \widetilde C|\over |\widetilde C|^{n-1\over n}}$$
is no larger than the corresponding expression for a regular $n$-dimensional ``tetrahedron''. It is also shown that among $n$-dimensional subspaces of $L_p$ (for each $p\in [1,\infty]), \ell^n_p$ has maximal volume ratio.\vskip3in
We prove a sharp isoperimetric inequality for the class of metric measure spaces verifying the synthetic Ricci curvature lower bounds $MCP(0,N)$ and having Euclidean volume growth at infinity. We avoid the classical use of the Brunn-Minkowski inequality, not available for $MCP(0,N)$, and of the PDE approach, not available in the singular setting. Our approach will be carried over by using a scaling limit of localisation.
In recent years, it has been shown that some classical inequalities follow from a local stochastic dominance for naturally associated random polytopes. We strengthen planar isoperimetric inequalities by attaching a stochastic model to some classical inequalities, such as Mahler's Theorem, and a reverse Lutwak-Zhang inequality, the polar for $L_p$ centroid bodies. In particular, we obtain the dual counterpart to a result of Bisztriczky-Böröczky.
It is shown that every even, zonal measure on the Euclidean unit sphere gives rise to an isoperimetric inequality for sets of finite perimeter which directly implies the classical Euclidean isoperimetric inequality. The strongest member of this large family of inequalities is shown to be the only affine invariant one among them - the Petty projection inequality. As an application, a family of sharp Sobolev inequalities for functions of bounded variation is obtained, each of which is stronger than the classical Sobolev inequality. Moreover, corresponding families of Lp isoperimetric and Sobolev…
We show a reverse isoperimetric inequality within the class of relative outer parallel bodies, with respect to a general convex body $E$, along with its equality condition. Based on the convexity of the sequence of quermassintegrals of Minkowski sums we also prove further inequalities.
We prove a reverse isoperimetric inequality for domains homeomorphic to a disc with the boundary of curvature bounded below lying in two-dimensional Alexandrov spaces of curvature $\geqslant c$. We also study the equality case.
We give a refinement of the quantitative isoperimetric inequality. We prove that the isoperimetric gap controls not only the Fraenkel asymmetry but also the oscillation of the boundary.
The isodiametric inequality is derived from the isoperimetric inequality trough a variational principle, establishing that balls maximize the perimeter among convex sets with fixed diameter. This principle brings also quantitative improvements to the isodiametric inequality, shown to be sharp by explicit nearly optimal sets.
We prove a sharp dimension-free isoperimetric inequality, involving the volume entropy, in non-compact metric measure spaces with non-negative synthetic Ricci curvature.
In this paper we prove an isoperimetric inequality of euclidean type for complete metric spaces admitting a cone-type inequality. These include all Banach spaces and all complete, simply-connected metric spaces of non-positive curvature in the sense of Alexandrov or, more generally, of Busemann. The main theorem generalizes results of Gromov and Ambrosio-Kirchheim.
Our main result gives an improved bound on the filling areas of closed curves in Banach spaces which are not closed geodesics. As applications we show rigidity of Pu's classical systolic inequality and investigate the isoperimetric constants of normed spaces. The latter has further applications concerning the regularity of minimal surfaces in Finsler manifolds.
M. Gromov introduced the Lipschitz order relation on the set of metric measure spaces and developed a rich theory. In particular, he claimed that an isoperimetric inequality on a non-discrete space is represented by using the Lipschitz order. We relax the definition of the Lipschitz order allowing an additive error to relate with an isoperimetric inequality on a discrete space. As an application, we obtain an isoperimetric inequality on the non-discrete $n$-dimensional $l^1$-cube by taking the limits of an isoperimetric inequality of the discrete $l^1$-cubes.
A purely analytic proof is given for an inequality that has as a direct consequence the two most important affine isoperimetric inequalities of plane convex geometry: The Blaschke-Santalo inequality and the affine isoperimetric inequality of affine differential geometry.
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…
We present a sharp version of the isoperimetric inequality for finitely generated groups due to T. Couhlon and L. Saloff-Coste based on the proof suggested by M. Gromov.
Let $G$ be a graph which satisfies $c^{-1} a^r \le |B(v,r)| \le c a^r$, for some constants $c,a>1$, every vertex $v$ and every radius $r$. We prove that this implies the isoperimetric inequality $|\partial A| \ge C |A| / \log(2+ |A|)$ for some constant $C=C(a,c)$ and every finite set of vertices $A$.
A stability version of the Blaschke-Santaló inequality and the affine isoperimetric inequality for convex bodies of dimension n>2 is proved. The first step is the reduction to the case when the convex body is o-symmetric and has axial rotational symmetry. This step works for related inequalities compatible with Steiner symmetrization. Secondly, for these convex bodies, a stability version of the characterization of ellipsoids by the fact that each hyperplane section is centrally symmetric is established.
In this paper we study the quantitative isoperimetric inequality in the plane. We prove the existence of a set $Ω$, different from a ball, which minimizes the ratio $δ(Ω)/λ^2(Ω)$, where $δ$ is the isoperimetric deficit and $λ$ the Fraenkel asymmetry, giving a new proof ofthe quantitative isoperimetric inequality. Some new properties of the optimal set are also shown.
We investigate isoperimetric constants of infinite tessellating metric graphs. We introduce a curvature-like quantity, which plays the role of a metric graph analogue of discrete curvature notions for combinatorial tessellating graphs. We then prove a lower estimate and a criterium for positivity of the isoperimetric constant.
We prove a pair of sharp reverse isoperimetric inequalities for domains in nonpositively curved surfaces: (1) metric disks centered at the vertex of a Euclidean cone of angle at least $2π$ have minimal area among all nonpositively curved disks of the same perimeter and the same total curvature; (2) geodesic triangles in a Euclidean (resp. hyperbolic) cone of angle at least $2π$ have minimal area among all nonpositively curved geodesic triangles (resp. all geodesic triangles of curvature at most $-1$) with the same side lengths and angles.
This paper announces the discovery of an isoperimetric inequality for the area of plane regions defined by binary forms. This result has been applied subsequently in the enumeration of solutions to the Thue inequality and, given its fundamental nature, may find application in other areas as well.
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…
In this paper we adapt the well-estabilished $Γ$-calculus techniques to the context of $RCD(K,\infty)$ spaces, proving Bobkov's local isoperimetric inequality and, when $K$ is positive, the Gaussian isoperimetric inequality in this class of spaces. The proof relies on the measure-valued $Γ_2$ operator introduced by Savaré.
We introduce tropical analogues of the notion of volume of polytopes, leading to a tropical version of the (discrete) classical isoperimetric inequality. The planar case is elementary, but a higher-dimensional generalization leads to an interesting class of ordinary convex polytopes, characterizing the equality case in the isoperimetric inequality. This study is motivated by open complexity questions concerning linear optimization and its tropical analogs.
We prove an isoperimetric inequality for the uniform measure on a uniformly convex body and for a class of uniformly log-concave measures (that we introduce). These inequalities imply (up to universal constants) the log-Sobolev inequalities proved by Bobkov--Ledoux as well as the isoperimetric inequalities due to Bakry-Ledoux and Bobkov--Zegarlinski. We also recover a concentration inequality for uniformly convex bodies, similar to that proved by Gromov--Milman.
We prove a discrete analogue to a classical isoperimetric theorem of Weil for surfaces with non-positive curvature. It is shown that hexagons in the triangular lattice have maximal volume among all sets of a given boundary in any triangulation with minimal degree 6.
Klartag's needle decomposition technique enables one to obtain strong isoperimetric inequalities on Riemannian manifolds other than the classical known examples. As a result, in this paper, we obtain sharp isoperimetric inequalities for compact rank one symmetric spaces (CROSS). Namely, for the real projective space $\mathbb{R}P^n$, we demonstrate that the isoperimetric regions are given by either the geodesic balls or tubes around some $\mathbb{R}P^k\subset\mathbb{R}P^n$. For the complex projective space $\mathbb{C}P^n$, the isoperimetric regions are given by either the geodesic balls or tube…
We prove that if $(X,\mathsf d,\mathfrak m)$ is an essentially non-branching metric measure space with $\mathfrak m(X)=1$, having Ricci curvature bounded from below by $K$ and dimension bounded from above by $N \in (1,\infty)$, understood as a synthetic condition called Measure-Contraction property, then a sharp isoperimetric inequality à la Lévy-Gromov holds true. Measure theoretic rigidity is also obtained.
We study harmonic and quasi-harmonic discs in metric spaces admitting a uniformly local quadratic isoperimetric inequality for curves. The class of such metric spaces includes compact Lipschitz manifolds, metric spaces with upper or lower curvature bounds in the sense of Alexandrov, some sub-Riemannian manifolds, and many more. In this setting, we prove local Hoelder continuity and continuity up to the boundary of harmonic and quasi-harmonic discs.
We establish a quantitative isoperimetric inequality for weighted Riemannian manifolds with $\mathrm{Ric}_{\infty} \ge 1$. Precisely, we give an upper bound of the volume of the symmetric difference between a Borel set and a sub-level (or super-level) set of the associated guiding function (arising from the needle decomposition), in terms of the deficit in Bakry-Ledoux's Gaussian isoperimetric inequality. This is the first quantitative isoperimetric inequality on noncompact spaces besides Euclidean and Gaussian spaces. Our argument makes use of Klartag's needle decomposition (also called local…
The notion of strictly outward minimising hull is investigated for open sets of finite perimeter sitting inside a complete noncompact Riemannian manifold. Under natural geometric assumptions on the ambient manifold, the strictly outward minimising hull $Ω^*$ of a set $Ω$ is characterised as a maximal volume solution of the least area problem with obstacle, where the obstacle is the set itself. In the case where $Ω$ has $\mathscr{C}^{1, α}$-boundary, the area of $\partial Ω^*$ is recovered as the limit of the $p$-capacities of $Ω$, as $p \to 1^+$. Finally, building on the existence of strictly …
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…
We obtain an explicit formula for comparing total curvature of level sets of functions on Riemannian manifolds, and develop some applications of this result to the isoperimetric problem in spaces of nonpositive curvature.
In this paper we study the relationship of hyperbolicity and (Cheeger) isoperimetric inequality in the context of Riemannian manifolds and graphs. We characterize the hyperbolic manifolds and graphs (with bounded local geometry) verifying this isoperimetric inequality, in terms of their Gromov boundary. Furthermore, we characterize the trees with isoperimetric inequality (without any hypothesis). As an application of our results, we obtain the solvability of the Dirichlet problem at infinity for these Riemannian manifolds and graphs, and that the Martin boundary is homeomorphic to the Gromov b…
The normalised volume measure on the $\ell_p^n$ unit ball ($1\leq p\leq 2$) satisfies the following isoperimetric inequality: the boundary measure of a set of measure $a$ is at least $cn^{1/p}\tilde{a}\log^{1-1/p}(1/\tilde{a})$, where $\tilde{a}=\min(a,1-a)$.
We prove a generalized isoperimetric inequality for a domain diffeomorphic to a sphere that replaces filling volume with $k$-dilation. Suppose $U$ is an open set in $\mathbb{R}^n$ diffeomorphic to a Euclidean $n$-ball. We show that in dimensions at least 4 there is a map from a standard Euclidean ball of radius about $vol(\partial U)^{1/(n-1)}$ to $U$, with degree 1 on the boundary, and $(n-1)$-dilation bounded by some constant only depending on $n$. We also give an example in dimension 3 of an open set where no such map with small $(n-1)$-dilation can be found. The generalized isoperimetric i…
We prove an inequality, valid on any finitely generated group with a fixed finite symmetric generating set, involving the growth of successive balls, and the average length of an element in a ball. It generalizes recent improvements of the Coulhon Saloff-Coste inequality. We reformulate the inequality in terms of the Følner function; in the case the finitely generated group is amenable with exponential growth, this allows us to express the best possible (outer) constant in the Coulhon Saloff-Coste isoperimetric inequality with the help of a formula involving the growth rate and the asymptotic …
We consider the family of $r$-parallel sets in $\mathbb{R}^d$, that is sets of the form $A_r=A+rB_2^n$, where $B_2^n$ is the unit Euclidean ball and $A$ is an arbitrary Borel set. We show that the ratio between the upper surface area measure of an $r$-parallel set and its volume is upper bounded by $d/r$. Equality is achieved for $A$ being a single point. As a consequence of our main result we show that the Gaussian upper surface area measure of an $r$-parallel set is upper bounded by $18d \max(\sqrt{d},r^{-1})$. Moreover, we observe that there exists a $1$-parallel set with Gaussian surface a…
We prove an inequality for submanifolds of Cartan-Hadamard manifolds, which relates the geometry of a submanifold to the measure of the geodesics in the ambient space which it intersects. For hypersurfaces, this gives an extension of Banchoff and Pohl's isoperimetric inequality to spaces of non-positive curvature. We also prove a modified version of Croke's isoperimetric inequality for hypersurfaces immersed in Cartan-Hadamard manifolds and a sharp, quantitative version of an isoperimetric inequality of Yau in spaces of negative curvature. We discuss the relationship between these results, and…
