Related papers: Metrics and Isometries for Convex Functions

We prove that a conformal mapping defined on the unit disk belongs to a weighted Bergman space if and only if certain integrals involving the harmonic measure converge. With the aid of this theorem, we give a geometric characterization of…

We study bounded domains $\Omega\subset\mathbb{C}^n$ whose Bergman metric is locally symmetric, i.e. its Riemannian curvature tensor is parallel with respect to the Levi-Civita connection. Following the strategy developed in…

We develop a notion of rank one properly convex domains (or Hilbert geometries) in the real projective space. This is in the spirit of rank one non-positively curved Riemannian manifolds and CAT(0) spaces. We define rank one isometries for…

The goal of this paper is to introduce and study analogues of the Euclidean Funk and Hilbert metrics on open convex subsets $\Omega$ of hyperbolic or spherical spaces. At least at a formal level, there are striking similarities among the…

In this paper we give some results about the approximation of a Lipschitz function on a Banach space by means of $\Delta$-convex functions. In particular, we prove that the density of $\Delta$-convex functions in the set of Lipschitz…

We prove that any metric with curvature $\leq -1$ (in the sense of A. D. Alexandrov) on a closed surface of genus $>1$ is isometric to the induced intrinsic metric on a space-like convex surface in a Lorentzian manifold of dimension $(2+1)$…

This paper focuses on estimating the Taylor coefficients for Hilbert spaces of holomorphic functions on the disk using intrinsic features of univalent functions and of Teichmuller spaces. Estimating these coefficients has a long history but…

In this note, we present two arguments showing that the classical \textit{linear adjoint cone restriction conjecture} holds for the class of functions supported on the cone and invariant under the spatial rotation in all dimensions. The…

In this paper, we study the Hausdorff dimension of self-similar measures and sets on the real line, where the generating iterated function system consists of some maps that share the same fixed point. In particular, we will show that out of…

A new nonparametric estimator of a convex regression function in any dimension is proposed and its convergence properties are studied. We start by using any estimator of the regression function and we \emph{convexify} it by taking the…

In this paper we introduce a new type of symmetrization, which preserves the anisotropic perimeter of the level sets of a suitable concave smooth function, in order to prove sharp comparison results for solutions of a class of homogeneous…

We explore the distinctions between $L^p$ convergence of metric tensors on a fixed Riemannian manifold versus Gromov-Hausdorff, uniform, and intrinsic flat convergence of the corresponding sequence of metric spaces. We provide a number of…

A Minkowski symmetral of an $\alpha$-concave function is introduced, and some of its fundamental properties are derived. It is shown that for a given $\alpha$-concave function, there exists a sequence of Minkowski symmetrizations that…

The goal of this work is to study the space of continuous functions whose ergodic averages converge everywhere towards a continuous function. We will connect, as in the case of a metric study, the convergence of the ergodic averages and the…

In this paper we study the idea of strong-I^K-convergence of functions which is common generalization of strong-I*-convergence of functions in probabilistic metric spaces. We also study strong-I^K-limit points of functions in the same…

We study the singularity (multifractal) spectrum of continuous convex functions defined on $[0,1]^{d}$. Let $E_f({h}) $ be the set of points at which $f$ has a pointwise exponent equal to $h$. We first obtain general upper bounds for the…

A classical result of variational analysis, known as Attouch theorem, establishes the equivalence between epigraphical convergence of a sequence of proper convex lower semicontinuous functions and graphical convergence of the corresponding…

An Ostrowski type integral inequality for convex functions and applications for quadrature rules and integral means are given. A refinement and a counterpart result for Hermite-Hadamard inequalities are obtained and some inequalities for…

We discuss five simple functions on finite multisets of metric spaces. The first four are all metrics iff the underlying space is bounded and are complete metrics iff it is also complete. Two of them, and the fifth function, all generalise…

We give precise estimates of some holomorphically invariant infinitesimal metrics near a pseudoconcave points in a wide family of ``model'' domains for that situation in $\mathbb C^2$. This extends to metrics (rather distances) the authors'…