Related papers: A weighted isoperimetric inequality on the hyperbo…

In this article we exhibit the largest constant in a quadratic isoperimetric inequality which ensures that a geodesic metric space is Gromov hyperbolic. As a particular consequence we obtain that Euclidean space is a borderline case for…

We consider the problem of minimizing the relative perimeter under a volume constraint in an unbounded convex body $C\subset \mathbb{R}^{n+1}$, without assuming any further regularity on the boundary of $C$. Motivated by an example of an…

We prove a Hardy-type inequality for the gradient of the Heisenberg Laplacian on open bounded convex polytopes on the first Heisenberg Group. The integral weight of the Hardy inequality is given by the distance function to the boundary…

We give a trigonometric proof of the Steiner-Lehmus Theorem in hyperbolic geometry. Precisely we show that if two internal bisectors of a triangle on the hyperbolic plane are equal, then the triangle is isosceles.

We provide several equivalent characterizations of Kobayashi hyperbolicity in unbounded convex domains in terms of peak and anti-peak functions at infinity, affine lines, Bergman metric and iteration theory.

In this paper we prove a general structure theorem for relatively hyperbolic groups (with arbitrary peripheral subgroups) acting naive convex co-compactly on properly convex domains in real projective space. We also establish a…

In this paper we address the reverse isoperimetric inequality for convex bodies with uniform curvature constraints in the hyperbolic plane $\mathbb{H}^2$. We prove that the\textit{ thick $\lambda$-sausage} body, that is, the convex domain…

We construct the hyperbolic plane with its geodesic flow as the scale plus symmetry reduction of a three-body problem in the Euclidean plane. The potential is $-I/\Delta^2$ where $I$ is the triangle's moment of inertia and $\Delta$ its…

For general varifolds in Euclidean space, we prove an isoperimetric inequality, adapt the basic theory of generalised weakly differentiable functions, and obtain several Sobolev type inequalities. We thereby intend to facilitate the use of…

A stability version of the reverse isoperimetric inequality, and the corresponding inequality for isotropic measures are established.

We generalize the hyperplane inequality in dimensions up to 4 to the setting of arbitrary measures in place of the volume. To prove this generalization we establish stability in the affirmative part of the solution to the Busemann-Petty…

The isoperimetric problem with a density or weighting seeks to enclose prescribed weighted volume with minimum weighted perimeter. According to Chambers' recent proof of the log-convex density conjecture, for many densities on…

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…

We prove an improved version of Poincar\'e-Hardy inequality in suitable subspaces of the Sobolev space on the hyperbolic space via Bessel pairs. As a consequence, we obtain a new Hardy type inequality with an improved constant (than the…

We compare and combine two approaches that have been recently introduced by Dafnis and Paouris [DP] and by Klartag and Milman [KM] with the aim of providing bounds for the isotropic constants of convex bodies. By defining a new hereditary…

We prove a sharp result for the distortion of a hyperbolic type metric under $K$-quasiregular mappings of the upper half plane. The proof makes use of a new kind of Bernoulli inequality and the Schwarz lemma for quasiregular mappings.

It is well known that a hyperbolic domain in the complex plane has uniformly perfect boundary precisely when the product of its hyperbolic density and the distance function to its boundary has a positive lower bound. We extend this…

The aim of this paper is to prove new trigonometric and hyperbolic inequalities, which constitute among others refinements or analogs of famous Cusa-Huygens, Wu-Srivastava, and related inequalities. In most cases, the obtained results are…

The discrete isoperimetric inequality states that among all n -gons with a fixed area, the regular n -gon has the least perimeter. We prove analogues of the discrete isoperimetric inequality (involving circumradius or inradius) for cyclic…

We prove an intrinsic equivalence between strong hypercontractivity and a strong logarithmic Sobolev inequality for the cone of logarithmically subharmonic functions. We introduce a new large class of measures, Euclidean regular and…