Related papers: On the Banach-Mazur Ellipse
The two-dimensional surface of a bi-axial ellipsoid is characterized by the lengths of its major and minor axes. Longitude and latitude span an angular coordinate system across. We consider the egg-shaped surface of constant altitude above…
The shortest distance between the first $n$ iterates of a typical point can be quantified with a log rule for some dynamical systems admitting Gibbs measures. We show this in two settings. For topologically mixing Markov shifts with at most…
In [1] new discretizations of the Euler top have been found. They can be discribed with a Lax pair with a spectral parameter on an elliptic curve. This is used in this paper to perform a finite gap integration.
We study the convergence of earthquake paths and horocycle paths in the Gardiner-Masur compactification of Teichm\"uller space. We show that an earthquake path directed by a uniquely ergodic or simple closed measured geodesic lamination…
We present an adaptation of the MA-LBR scheme to the Monge-Amp{\`e}re equation with second boundary value condition, provided the target is a convex set. This yields a fast adaptive method to numerically solve the Optimal Transport problem…
We prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth $3$-dimensional Riemannian manifolds. The constructed min-max surfaces are smooth with at…
We report on measurements performed on an apparatus aimed to study the Casimir force in the cylinder-plane configuration. The electrostatic calibrations evidence anomalous behaviors in the dependence of the electrostatic force and the…
The classical Mazur-Ulam theorem establishes that every surjective isometry between normed real vector spaces is an affine transformation. In various applied mathematical settings, however, one encounters maps that preserve distances not…
We study $C^2$ extremal quasiconformal mappings in space and establish necessary and sufficient conditions for a `localized' form of extremality in the spirit of the work of G. Aronsson on absolutely minimizing Lipschitz extensions. We also…
We study shortest paths and their distances on a subset of a Euclidean space, and their approximation by their equivalents in a neighborhood graph defined on a sample from that subset. In particular, we recover and extend the results of…
A compactness theorem for volume-constrained almost-critical points of elliptic integrands is proven. The result is new even for the area functional, as almost-criticality is measured in an integral rather than in a uniform sense. Two main…
We investigate minimal-perimeter configurations of two finite sets of points on the square lattice. This corresponds to a lattice version of the classical double-bubble problem. We give a detailed description of the fine geometry of…
We generalize the Arzel\`a-Ascoli theorem in the space of continuous maps on a compact interval with values in Euclidean N-space by providing a quantitative link between the Hausdorff measure of noncompactness in this space and a natural…
The aim of this paper is to present two tools, Theorems 4 and 7, that make the task of finding equivalent polyhedral norms on certain Banach spaces easier and more transparent. The hypotheses of both tools are based on countable…
In 2022, Hatori gave a sufficient condition for complex Banach spaces to have the complex Mazur--Ulam property. In this paper, we introduce a class of complex Banach spaces $B$ that do not satisfy the condition but enjoy the property that…
We propose a simple derivation of an upper bound for the perimeter of an ellipse. The procedure, which relies on the use of elliptic integrals, consists in introducing, via inequalities and convexity properties, specific integrals which can…
This paper establishes a new Chebyshev criterion for some family of integrals. By virtue of this criterion we obtain several new Chebyshev families. With the help of these new families we can answer the conjecture posed by Gasull et al in…
We give a formalism for approximate isomorphism in continuous logic simultaneously generalizing those of two papers by Ben Yaacov and by Ben Yaacov, Doucha, Nies, and Tsankov, which are largely incompatible. With this we explicitly exhibit…
We show that all the standard distances from metric geometry and functional analysis, such as Gromov-Hausdorff distance, Banach-Mazur distance, Kadets distance, Lipschitz distance, Net distance, and Hausdorff-Lipschitz distance have all the…
We analyse several examples of separable Banach spaces, some of them new, and relate them to several dichotomies obtained in the previous paper Banach spaces without minimal subspaces, by classifying them according to which side of the…