Abstract Quasi-isometries are a versatile type of maps that preserve the large-scale geometry of spaces, while introducing significant local distortions. Following Kanai’s work, which established the invariance of various analytic and geometric properties under quasi-isometries, this paper generalizes isoperimetric and Sobolev inequalities for exponents less than the manifold’s dimension, proving both that they are equivalent and preserved by quasi-isometries.

Abstract The convexity analysis of elliptic partial differential equations originates from the natural phenomenon modeling in geometry and physics, such as minimal surfaces, isoperimetric problems, and optimal transport. In recent years, the research on the geometric properties of solutions to nonlinear equations, such as p-harmonic functions and k-Hessian equations, has made significant progress. Convexity is one of the most representative geometric characteristics, and curvature estimates are one of the common methods for quantitative research. This paper mainly studies two types of Monge-Ampere equations and Hessian equations with 0 boundary value Dirichlet problems. It obtains differential inequalities related to the curvature of the level sets of the solutions, ensuring that the operator acting on the function is greater than or equal to zero. Furthermore, based on the principle of extremum, the function reaches its maximum value at the boundary, and the curvature estimate of the solution level set is ultimately obtained.

Abstract The paper is devoted to proving Allard–Michael–Simon‐type ‐Sobolev inequalities with explicit constants in the setting of Euclidean minimal submanifolds of arbitrary codimension. Our results require separate discussions for the cases and , respectively. In particular, for , we obtain an asymptotically sharp and codimension‐free Sobolev constant. Our argument is based on the optimal mass transport theory on Euclidean submanifolds and also provides an alternative, unified proof of the recent isoperimetric inequalities of Brendle (J. Amer. Math. Soc., 2021) and Brendle and Eichmair (Notices Amer. Math. Soc., 2024).

The Berry phase and quantum distance characterize the geometry of wave functions in Hilbert space. We uncover a quantum analog of the classical isoperimetric inequality, revealing a fundamental link between the Berry phase and quantum distance along closed paths. For two-band systems, we establish a strong inequality analogous to the spherical isoperimetric problem. For multiband systems, we prove a weak inequality showing that the Berry phase never exceeds the quantum distance. These results introduce new quantum geometric principles, placing fundamental bounds on physical quantities such as the Wannier function spread, quantum speed, electron-phonon coupling, and superfluid weight, with broad implications across condensed matter, quantum information, and beyond.

Abstract To each finitely generated group G , we associate a quasi-isometric invariant called the Dehn spectrum of G . If G is finitely presented, our invariant is closely related to the Dehn function of G , yet provides more information by encoding the isoperimetric behaviour of G at various scales. The main goal of this paper is to initiate the study of the Dehn spectrum of finitely generated (but not necessarily finitely presented) groups. In particular, we compute the Dehn spectrum of small cancellation groups, certain wreath products, and free Burnside groups of sufficiently large odd exponent. We also address some natural questions on the structure of the poset of Dehn spectra. As an application, we show that there exist $$2^{\aleph _0}$$ 2 ℵ 0 pairwise non-quasi-isometric finitely generated groups of finite exponent.

We study the torsional rigidity for rods with thinning cross sections. The main purpose of the paper is to prove rigorous asymptotic formulas for the torsional rigidity as the thickness of the cross section tends to zero. These asymptotic formulas are empirically known and are widely used in the field of Mechanics of Materials. From a more theoretical point of view, thinning domains are considered when studying optimal inequalities for suitable classes of functionals depending on domains. We recall as an example of this kind of inequalities the celebrated Saint Venant inequality stating that, among planar domains with fixed Lebesgue measure, the disk is the cross section corresponding to a maximal torsional rigidity. It is well known that this statement is equivalent to say that disks are maximizer of a suitable functional, see for example (Amato et al. in On the optimal sets in Pólya and Makai type inequalities, 2025) and the references therein. Actually, in the present paper other kinds of functionals are more relevant when considering thinning domains. We refer in particular to the so-called Pólya and Makai functionals, see the papers (Makai in On the principal frequency of a membrane and the torsional rigidity of a beam, Stanford Univ. Press, Stanford, 1962; Pólya, J Indian Math Soc (N.S.) 24(1960):413–419, 1961; Pólya and Szegö, Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, Princeton, 1951) for more details and (1.14) in the present paper for the precise definitions.

Reliability analysis of the quinary n-cube networks with non-lexicographic order optimal solution of the edge isoperimetric problem

In the present manuscript, we address and solve for the first time a nonlocal discrete isoperimetric problem. We consider indeed a generalization of the classical perimeter, what we call a nonlocal bi-axial discrete perimeter , where not only the external boundary of a polyomino \mathcal{P} contributes to the perimeter, but all internal and external components of \mathcal{P} . Furthermore, we find and characterize its minimizers in the class of polyominoes with fixed area n . Moreover, we explain how the solution of the nonlocal discrete isoperimetric problem is related to the rigorous study of the metastable behavior of a long-range bi-axial Ising model .

In this paper, we study thermodynamics and its applications of a family of static charged dilaton black holes in 2+1 dimensions found by Chan and Mann [1] and Xu [2]. There is a dimensionless parameter [Formula: see text] in the black hole solutions presented: it is related to the coupling constant for the dilaton with the electromagnetic field and the gravitational field. Black hole horizons exist only for [Formula: see text]. [Formula: see text] black hole is a solution to low energy string theory. Thermodynamics is studied in the canonical ensemble where charge is constant as well as in grand canonical ensemble where the potential is constant. The cosmological constant is considered as a thermodynamical variable where the pressure [Formula: see text]. We computed the first law and the Smarr relations for the black hole and introduced two new thermodynamical parameter in order to satisfy the first law. We computed temperature, thermodynamic volume, specific heat capacities, Gibbs free energy and studied local and global stability of the black hole. Thermodynamic volume differs from the geometric volume. In the canonical ensemble, we noticed that thermodynamic behavior falls into two broad categories: For [Formula: see text], small black holes are locally stable and large black holes are not. For [Formula: see text] the black hole is locally and globally stable for all values of the horizon radius. In order to demonstrate the two broad categories, we have presented [Formula: see text] and [Formula: see text] black holes in detail. There were no phase transitions for the above values of [Formula: see text]. In the grand canonical ensemble, we noticed that there is a Hawking-Page phase transition for the black hole with [Formula: see text]. We have also studied the Joule-Thomson expansion and the Reverse Isoperimetric Inequality of these black holes. We made the observation that the charged dilaton black hole does not violate the Reverse Isoperimetric Inequality for certain values of the parameters of the theory. Finally, we have suggested future work.

Abstract We extend Groman and Solomon's reverse isoperimetric inequality to pseudoholomorphic curves with punctures at the boundary and whose boundary components lie in a collection of Lagrangian submanifolds with intersections locally modelled on inside . Our construction closely follows the methods used by Duval and Abouzaid and corrects an error appearing in the latter approach.

Abstract We resolve the static isoperimetric problem underlying the Mandelstam–Tamm limit: among one-dimensional confining potentials at fixed gap Δ, the harmonic trap uniquely maximizes Var 0 ( x ) , yielding the exact geometric quantum speed limit g x x ⩽ ( 2 m Δ ) − 1 with an iff criterion. Beyond the extremum we prove quantitative rigidity (Thomas–Reiche–Kuhn-tail and structural L 2 control), extend to magnetic settings (longitudinal iff; transverse guiding-center bounds), and note applications to static polarizability, quantum-metric limits, and trap benchmarking.

In this paper, we establish higher-order Sobolev and Rellich-type inequalities on non-compact Riemannian manifolds supporting an isoperimetric inequality. We highlight two notable settings: manifolds with non-negative Ricci curvature and having Euclidean volume growth (supporting Brendle's isoperimetric inequality), and manifolds with non-positive sectional curvature (satisfying the Cartan–Hadamard conjecture or supporting Croke's isoperimetric inequality). Our proofs rely on various symmetrization techniques, and the key ingredient is an iterated Talenti's comparison principle. The non-iterated version is analogous to the main result of Chen and Li [J. Geom. Anal., 2023].

An affine isoperimetric inequality for log-concave functions

The isoperimetric problem for hyperideal polyhedra, part II

In this paper, we take a simplicial complex of dimension [Formula: see text] and we introduce on it oriented [Formula: see text]-simplexes and oriented [Formula: see text]-simplexes. As a result, we create a weighted simplicial set of [Formula: see text]-simplexes and [Formula: see text]-simplexes [Formula: see text], on which we introduce the cochains of dimension [Formula: see text] and we use them to construct a weighted normalized Laplacian associated with [Formula: see text]-simplexes [Formula: see text] and its weighted Dirichlet normalized Laplacian associated with [Formula: see text]-simplexes [Formula: see text]. Next, we study the spectrum of [Formula: see text] and the spectrum of [Formula: see text]. Finally, we develop the weighted isoperimetric inequalities at infinity associated to [Formula: see text] and we use them to analyze the essential spectrum of [Formula: see text] and the essential spectrum of [Formula: see text].

The solar-periodic system is a completely self-contained space, with no singularity or boundaries, which timelessly regenerates preserving its initial states (origins) and completely described by Euler’s complex theory of holomorphic (complex-smooth-recurrent) functions, ever discovered. This unified theory is the basis of eternal existence furnishing the building blocks of reality and the solution for long-standing problem of Cartesian dualism. It demonstrates precisely how mind generates relative/temporary matter in a metastable (phasing) equilibrium through the quantum dual isomorphism (e, p) of light, mutually regenerating matter. Physically it manifests as the quantum recurrence at the three levels/scales: quantum chemical elements (Mendeleev’s law, atomic frozen spiral), molecular thermistors (thermal molecular spiral) and astronomical gravitational rotating bodies (global/integrated solar system of planets and satellites). The periodical chemical elements are the first 94 known to occur naturally on Earth, of which 36 are primordial making up “the permafrost”. These are elements from the p-block (including hydrogen and helium) of the periodic table, where their atomic structure has completely filled subshells preserving initial states are daily and monthly (Krypton) regenerating (24 hours-one day) ≅1 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑚 𝑚𝑚𝑚𝑚𝑚𝑚 (𝑃𝑃𝑢𝑢)𝑔𝑔0 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑛𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛𝑛𝑛 (𝑍𝑍). They genuinely have a dual character (two allotropes/states), owing to the primal quantum contact ((𝑒𝑒+𝑒𝑒2)1/4≡𝜋𝜋1/2). The molecular structures and astronomical body configurations, self-contained them, are regenerating after a year and respectively century following a binary gravitational rule, 􁉀𝑔𝑔022􁉁12=7 (𝑙𝑙𝑙𝑙𝑙𝑙𝑔 0=1), for critical mixtures, the so-called the relative peakedness or two soliton coherent distribution (phases). The solar regenerative system is a critical system in the sense that is “an autocatalytic thermomolecular reaction” regenerating its initial states only in strict conditions. In such a system the reaction products increase the rate of reaction and if the ratio of system surface to system volume is large, then the reaction products tend to escape at the boundaries of system. Contrary, if the surface to volume ratio is small then the rate of escape may be less than the rate of formation and the reaction rate may extinguish or re-ignite for a critical size where rate of production just equals the rate of removal. At such a critical size, given by Sobolev isoperimetric inequality (4𝜋𝜋𝜋𝜋 (𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎)≤𝑃𝑃2 (𝑝𝑝𝑝 𝑝 𝑝𝑝𝑝 𝑝𝑝𝑝 )), the reaction is self-sustaining and is going on for so-long time the rate of regeneration does not exceed the rate of production. The bases of the Eulerian quantum dual isomorphism of light which explain close relationship between quantum automorphic wary light and the solar timelessly regenerative system, in a photostationary (or frozen) equilibrium, are considered.

A major research area in discrete geometry is to consider the best way to partition the d-dimensional Euclidean space mathbb {R}^d under various quality criteria. In this paper we introduce a new type of space partitioning that is motivated by the problem of rounding noisy measurements from the continuous space mathbb {R}^d to a discrete subset of representative values. Specifically, we study partitions of mathbb {R}^d into bounded-size tiles colored by one of k colors, such that tiles of the same color have a distance of at least t from each other. Such tilings allow for error-resilient rounding, as two points of the same color and distance less than t from each other are guaranteed to belong to the same tile, and thus, to be rounded to the same point. The main problem we study in this paper is characterizing the achievable tradeoffs between the number of colors k and the distance t, for various dimensions d. On the qualitative side, we show that in mathbb {R}^d, using k=d+1 colors is both sufficient and necessary to achieve t>0. On the quantitative side, we achieve numerous upper and lower bounds on t as a function of k. In particular, for d=3,4,8,24, we obtain sharp asymptotic bounds on t, as k rightarrow infty . We obtain our results with a variety of techniques including isoperimetric inequalities, the Brunn-Minkowski theorem, sphere packing bounds, Bapat’s connector-free lemma, and Čech cohomology.

Abstract We study isoperimetric inequalities on “slabs”, namely weighted Riemannian manifolds obtained as the product of the uniform measure on a finite length interval with a codimension‐one base. As our two main applications, we consider the case when the base is the flat torus and the standard Gaussian measure on . The isoperimetric conjecture on the three‐dimensional cube predicts that minimizers are enclosed by spheres about a corner, cylinders about an edge and coordinate planes. This has only been established for relative volumes close to 0, and 1 by compactness arguments. Our analysis confirms the isoperimetric conjecture on the three‐dimensional cube with side lengths in a new range of relative volumes . In particular, we confirm the conjecture for the standard cube () for all , when for the entire range where spheres are conjectured to be minimizing, and also for all . When we reduce the validity of the full conjecture to establishing that the half‐plane is an isoperimetric minimizer. We also show that the analogous conjecture on a high‐dimensional cube is false for . In the case of a slab with a Gaussian base of width , we identify a phase transition when and when . In particular, while products of half‐planes with are always minimizing when , when they are never minimizing, being beaten by Gaussian unduloids. In the range , a potential trichotomy occurs.

A bstract We investigate the thermodynamic and holographic properties of charged and rotating quantum black holes in a doubly holographic braneworld setup. These quantum black holes are derived from the anti-de Sitter C-metric and are exact solutions to a semiclassical gravitational theory which incorporates all orders of the backreaction of quantum fields on spacetime. The inclusion of both charge and rotation extends and generalizes previous studies. The thermodynamics and critical behavior of the black holes are examined from the bulk, brane, and boundary perspectives, and we demonstrate that the inclusion of either charge or rotation removes the reentrant phase transitions seen in the neutral-static case. The critical exponents of the system are calculated using numerical methods and found to differ from the standard mean field theory values for the neutral-static black holes’ reentrant phase transitions, but in agreement with mean-field theory for the phase transitions of the black holes with charge and rotation. Additionally, to test the validity of the semiclassical treatment, we study a mass-gap energy scale to identify regimes where quantum fluctuations of spacetime geometry are expected to become significant and speculate about a connection with weak cosmic censorship gedankenexperiments. We also generalize the quantum Penrose inequality and the quantum reverse isoperimetric inequality to include charge and rotation. Finally, we compute a renormalized gyromagnetic ratio and analyze it in the limit of large backreaction.

The projection entropy of convex bodies in [Formula: see text] is introduced. By using introduced [Formula: see text]-quermassintegrals, we prove an isoperimetric-type inequality for projection entropy. Some stronger inequalities than the classical isoperimetric inequality and the dual Urysohn inequality are obtained.