Reverse Isoperimetric Inequality Research Articles

Bonnesen-style Wulff isoperimetric inequality

The Wulff isoperimetric inequality is a natural extension of the classical isoperimetric inequality (Green and Osher in Asian J. Math. 3:659-676 1999). In this paper, we establish some Bonnesen-style Wulff isoperimetric inequalities and reverse Bonnesen-style Wulff isoperimetric inequalities. Those inequalities obtained are extensions of known Bonnesen-style inequalities and reverse Bonnesen-style inequalities.

Thermodynamic volume of cosmological solitons

We present explicit expressions of the thermodynamic volume inside and outside the cosmological horizon of Eguchi-Hanson solitons in general odd dimensions. These quantities are calculable and well-defined regardless of whether or not the regularity condition for the soliton is imposed. For the inner case, we show that the reverse isoperimetric inequality is not satisfied for general values of the soliton parameter $a$, though a narrow range exists for which the inequality does hold. For the outer case, we find that the mass $M_{out}$ satisfies the maximal mass conjecture and the volume is positive. We also show that, by requiring $M_{out}$ to yield the mass of dS spacetime when the soliton parameter vanishes, the associated cosmological volume is always positive.

BRASCAMP–LIEB INEQUALITY AND QUANTITATIVE VERSIONS OF HELLY'S THEOREM

We provide new quantitative versions of Helly's theorem. For example, we show that for every family of closed half-spaces in such that has positive volume, there exist and such that where are absolute constants. These results complement and improve previous work of Bárány et al and Naszódi. Our method combines the work of Srivastava on approximate John's decompositions with few vectors, a new estimate on the corresponding constant in the Brascamp–Lieb inequality and an appropriate variant of Ball's proof of the reverse isoperimetric inequality.

Volume inequalities of convex bodies from cosine transforms on Grassmann manifolds

The Lp cosine transform on Grassmann manifolds naturally induces finite dimensional Banach norms whose unit balls are origin-symmetric convex bodies in Rn. Reverse isoperimetric type volume inequalities for these bodies are established, which extend results from the sphere to Grassmann manifolds.

Inequalities for hyperconvex sets

Abstract An r-hyperconvex body is a set in the d-dimensional Euclidean space 𝔼 d that is the intersection of a family of closed balls of radius r. We prove the analogue of the classical Blaschke–Santaló inequality for r-hyperconvex bodies, and we also establish a stability version of it. The other main result of the paper is an r-hyperconvex version of the reverse isoperimetric inequality in the plane.

Relative Stanley–Reisner theory and Upper Bound Theorems for Minkowski sums

In this paper we settle two long-standing questions regarding the combinatorial complexity of Minkowski sums of polytopes: We give a tight upper bound for the number of faces of a Minkowski sum, including a characterization of the case of equality. We similarly give a (tight) upper bound theorem for mixed facets of Minkowski sums. This has a wide range of applications and generalizes the classical Upper Bound Theorems of McMullen and Stanley. Our main observation is that within (relative) Stanley–Reisner theory, it is possible to encode topological as well as combinatorial/geometric restrictions in an algebraic setup. We illustrate the technology by providing several simplicial isoperimetric and reverse isoperimetric inequalities in addition to our treatment of Minkowski sums.

The Minimal Dual Orlicz Surface Area

Petty proved that a convex body in $\mathbb{R}^{n}$ has the minimal surface area among its $\operatorname{SL}(n)$ images if and only if its surface area measure is isotropic. Recently, Zou and Xiong generalized this result to the Orlicz setting by introducing a new notion of minimal Orlicz surface area, and the analog of Ball's reverse isoperimetric inequality is established. In this paper, we give the dual results in Orlicz space by introducing a new notion of minimal dual Orlicz surface area. And the dual form of Ball's isoperimetric inequality is established.

Lower-dimensional black hole chemistry

The connection between black hole thermodynamics and chemistry is extended to the lower-dimensional regime by considering the rotating and charged BTZ metric in the $(2+1)$-D and a $(1+1)$-D limits of Einstein gravity. The Smarr relation is naturally upheld in both BTZ cases, where those with $Q \ne 0$ violate the Reverse Isoperimetric Inequality and are thus superentropic. The inequality can be maintained, however, with the addition of a new thermodynamic work term associated with the mass renormalization scale. The $D\rightarrow 0$ limit of a generic $D+2$-dimensional Einstein gravity theory is also considered to derive the Smarr and Komar relations, although the opposite sign definitions of the cosmological constant and thermodynamic pressure from the $D>2$ cases must be adopted in order to satisfy the relation. The requirement of positive entropy implies a lower bound on the mass of a $(1+1)$-D black hole. Promoting an associated constant of integration to a thermodynamic variable allows one to define a "rotation" in one spatial dimension. Neither the $D=3$ nor the $D \rightarrow 2$ black holes exhibit any interesting phase behaviour.

Entropy Inequality Violations from Ultraspinning Black Holes.

We construct a new class of rotating anti-de Sitter (AdS) black hole solutions with noncompact event horizons of finite area in any dimension and study their thermodynamics. In four dimensions these black holes are solutions to gauged supergravity. We find that their entropy exceeds the maximum implied from the conjectured reverse isoperimetric inequality, which states that for a given thermodynamic volume, the black hole entropy is maximized for Schwarzschild-AdS space. We use this result to suggest more stringent conditions under which this conjecture may hold.

On a result by Y. Groman and J. P. Solomon

We give a short proof of a reverse isoperimetric inequality due to Y. Groman and J. P. Solomon.

Ultraspinning limits and super-entropic black holes

By employing the new ultraspinning limit we construct novel classes of black holes with non-compact event horizons and finite horizon area and study their thermodynamics. Our ultraspinning limit can be understood as a simple generating technique that consists of three steps: i) transforming the known rotating AdS black hole solution to a special coordinate system that rotates (in a given 2-plane) at infinity ii) boosting this rotation to the speed of light iii) compactifying the corresponding azimuthal direction. In so doing we qualitatively change the structure of the spacetime since it is no longer possible to return to a frame that does not rotate at infinity. The obtained black holes have non-compact horizons with topology of a sphere with two punctures. The entropy of some of these exceeds the maximal bound implied by the reverse isoperimetric inequality, such black holes are super-entropic.

Isomorphic versions of reverse isoperimetric inequalities

Isomorphic versions of the \(L_p\) volume ratio inequalities are established. For the asymmetric cases, using the approach of isotropic embeddings, we also obtain isomorphic versions of the Ball–Barthe simplex inequalities.

On the shape of a convex body with respect to its second projection body

We prove results relative to the problem of finding sharp bounds for the affine invariant P(K)=V(ΠK)/Vd−1(K). Namely, we prove that if K is a 3-dimensional zonoid of volume 1, then its second projection body Π2K is contained in 8K, while if K is any symmetric 3-dimensional convex body of volume 1, then Π2K contains 6K. Both inclusions are sharp. Consequences of these results include a stronger version of a reverse isoperimetric inequality for 3-dimensional zonoids established by the author in a previous work, a reduction for the 3-dimensional Petty conjecture to another isoperimetric problem and the best known lower bound up to date for P(K) in 3 dimensions. As byproduct of our methods, we establish an almost optimal lower bound for high-dimensional bodies of revolution.

Affine images of isotropic measures

Necessary and sufficient conditions are given in order for a Borel measure on the Euclidean sphere to have an affine image that is isotropic. A sharp reverse affine isoperimetric inequality for Borel measures on the sphere is presented. This leads to sharp reverse affine isoperimetric inequalities for convex bodies. © 2015, International Press of Boston, Inc. All rights reserved.

On reverse isoperimetric inequalities in two-dimensional space forms and related results

On reverse isoperimetric inequalities in two-dimensional space forms and related results

Charged de Sitter-like black holes: quintessence-dependent enthalpy and new extreme solutions

We consider Reissner–Nordström black holes surrounded by quintessence where both a non-extremal event horizon and a cosmological horizon exist besides an inner horizon ( $$-1\le \omega <-1/3$$ ). We determine new extreme black hole solutions that generalize the Nariai horizon to asymptotically de Sitter-like solutions for any order relation between the squares of the charge $$q^2$$ and the mass parameter $$M^2$$ provided $$q^2$$ remains smaller than some limit, which is larger than $$M^2$$ . In the limit case $$q^2=9\omega ^2 M^2/(9\omega ^2-1)$$ , we derive the general expression of the extreme cosmo-black-hole, where the three horizons merge, and we discuss some of its properties. We also show that the endpoint of the evaporation process is independent of any order relation between $$q^2$$ and $$M^2$$ . The Teitelboim energy and the Padmanabhan energy are related by a nonlinear expression and are shown to correspond to different ensembles. We also determine the enthalpy $$H$$ of the event horizon, as well as the effective thermodynamic volume which is the conjugate variable of the negative quintessential pressure, and show that in general the mass parameter and the Teitelboim energy are different from the enthalpy and internal energy; only in the cosmological case, that is, for Reissner–Nordström–de Sitter black hole we have $$H=M$$ . Generalized Smarr formulas are also derived. It is concluded that the internal energy has a universal expression for all static charged black holes, with possibly a variable mass parameter, but it is not a suitable thermodynamic potential for static-black-hole thermodynamics if $$M$$ is constant. It is also shown that the reverse isoperimetric inequality holds. We generalize the results to the case of the Reissner–Nordström–de Sitter black hole surrounded by quintessence with two physical constants yielding two thermodynamic volumes.

The minimal Orlicz surface area

Petty proved that a convex body in Rn has the minimal surface area amongst its SL(n) images, if, and only if, its surface area measure is isotropic. By introducing a new notion of minimal Orlicz surface area, we generalize this result to the Orlicz setting. The analog of Ball's reverse isoperimetric inequality is established.

A reverse isoperimetric inequality for J-holomorphic curves

We prove that the length of the boundary of a $J$-holomorphic curve with Lagrangian boundary conditions is dominated by a constant times its area. The constant depends on the symplectic form, the almost complex structure, the Lagrangian boundary conditions and the genus. A similar result holds for the length of the real part of a real $J$-holomorphic curve. The infimum over $J$ of the constant properly normalized gives an invariant of Lagrangian submanifolds. We calculate this invariant to be $2\pi$ for the Lagrangian submanifold $\mathbb R P^n \subset \mathbb C P^n.$ We apply our result to prove compactness of moduli of $J$-holomorphic maps to non-compact target spaces that are asymptotically exact. In a different direction, our result implies the adic convergence of the superpotential.

Convex ancient solutions of the mean curvature flow

We study solutions of the mean curvature flow which are defined for all negative curvature times, usually called ancient solutions. We give various conditions ensuring that a closed convex ancient solution is a shrinking sphere. Examples of such conditions are: a uniform pinching condition on the curvatures, a suitable growth bound on the diameter or a reverse isoperimetric inequality. We also study the behaviour of uniformly k-convex solutions, and consider generalizations to ancient solutions immersed in a sphere.

Thermodynamics of Rotating Black Holes and Black Rings: Phase Transitions and Thermodynamic Volume

In this review we summarize, expand, and set in context recent developments on the thermodynamics of black holes in extended phase space, where the cosmological constant is interpreted as thermodynamic pressure and treated as a thermodynamic variable in its own right. We specifically consider the thermodynamics of higher-dimensional rotating asymptotically flat and AdS black holes and black rings in a canonical (fixed angular momentum) ensemble. We plot the associated thermodynamic potential—the Gibbs free energy—and study its behavior to uncover possible thermodynamic phase transitions in these black hole spacetimes. We show that the multiply-rotating Kerr-AdS black holes exhibit a rich set of interesting thermodynamic phenomena analogous to the “every day thermodynamics” of simple substances, such as reentrant phase transitions of multicomponent liquids, multiple first-order solid/liquid/gas phase transitions, and liquid/gas phase transitions of the van derWaals type. Furthermore, the reentrant phase transitions also occur for multiply-spinning asymptotically flat Myers–Perry black holes. These phenomena do not require a variable cosmological constant, though they are more naturally understood in the context of the extended phase space. The thermodynamic volume, a quantity conjugate to the thermodynamic pressure, is studied for AdS black rings and demonstrated to satisfy the reverse isoperimetric inequality; this provides a first example of calculation confirming the validity of isoperimetric inequality conjecture for a black hole with non-spherical horizon topology. The equation of state P = P(V,T) is studied for various black holes both numerically and analytically—in the ultraspinning and slow rotation regimes.