arXiv subjects

Ralph Teixeira

Publications and source records attributed to Ralph Teixeira.

17 recordsLinked to original sources

arXiv2019

Convex analysis in normed spaces and metric projections onto convex bodies

We investigate metric projections and distance functions referring to convex bodies in finite-dimensional normed spaces. For this purpose we identify the vector space with its dual space by using, instead of the usual identification via the standard inner product, the Legendre transform associated with the given norm. This approach yields re-interpretations of various properties of convex functions, and new relations between such functions and geometric properties of the studied norm are also derived.

math.MG↗

arXiv2019

Asymmetry measures for convex distance functions

Gauges, or convex distance functions are, roughly speaking, norms without symmetry. In this paper we intend to quantify how asymmetric a planar gauge can be. We introduce asymmetry measures for smooth gauges and for strictly convex gauges, prove that they are invariant under isometries, and investigate lower and upper bounds for them. Identifying a gauge with a convex body containing the origin in its interior (the unit ball of the gauge), we also prove that all introduced asymmetry measures are continuous in the Hausdorff distance. Finally, we show that, modifying one of the constructed asymmetry measures, a certain duality principle holds.

math.MG↗

arXiv2019

Duality of gauges and symplectic forms in vector spaces

A gauge $γ$ in a vector space $X$ is a distance function given by the Minkowski functional associated to a convex body $K$ containing the origin in its interior. Thus, the outcoming concept of gauge spaces $(X, γ)$ extends that of finite dimensional real Banach spaces by simply neglecting the symmetry axiom (a viewpoint that Minkowski already had in mind). If the dimension of $X$ is even, then the fixation of a symplectic form yields an identification between $X$ and its dual space $X^*$ . The image of the polar body $K^{\circ}\subseteq X^*$ under this identification yields a (skew-)dual gauge on $X$. In this paper, we study geometric properties of this so-called dual gauge, such as its behavior under isometries and its relation to orthogonality. A version of the Mazur-Ulam theorem for gauges is also proved. As an application of the theory, we show that closed characteristics of the boundary of a (smooth) convex body are optimal cases of a certain isoperimetric inequality.

math.MG↗

arXiv2018

A new geometric viewpoint on Sturm-Liouville eigenvalue problems

In Euclidean plane geometry, cycloids are curves which are homothetic to their respective bi-evolutes. In smooth normed planes, cycloids can be similarly defined, and they are characterized by their radius of curvature functions being solutions to eigenvalue problems of certain Sturm-Liouville equations. In this paper, we show that, for the eigenvalue λ = 1, this equation is a previously studied Hill equation which is closely related to the geometry given by the norm. We also investigate which geometric properties can be derived from this equation. Moreover, we prove that if the considered norm is defined by a Radon curve, then the solutions to the Hill equation are given by trigonometric functions. Further, we give conditions under which a given Hill equation induces a planar Minkowski geometry, and we prove that in this case the geometry is Euclidean if an eigenvalue other than λ = 1 induces a reparametrization of the original unit circle.

math.CA↗

arXiv2018

On curvature of surfaces immersed in normed spaces

The normal map given by Birkhoff orthogonality yields extensions of principal, Gaussian and mean curvatures to surfaces immersed in three-dimensional spaces whose geometry is given by an arbitrary norm and which are also called Minkowski spaces. We obtain characterizations of the Minkowski Gaussian curvature in terms of surface areas, and respective generalizations of the classical theorems of Huber, Willmore, Alexandrov, and Bertrand-Diguet-Puiseux are derived. A generalization of Weyl's formula for the volume of tubes and some estimates for volumes and areas in terms of curvature are obtained, and in addition we discuss also two-dimensional subcases of the results in more detail.

math.DG↗

arXiv2017

Optimal constants in normed planes

We define new geometric constants for normed planes, determine their optimal values, and characterize types of planes for which these optimal values are attained. Relations of these constants to several topics, such as areas and distances from points to sides of triangles, are also investigated. We perform some calculations of new and old constants given by trigonometric functions for certain classes of norms, and a conjecture on vectors which are isosceles orthogonal is disproved.

math.MG↗

arXiv2017

Differential geometry of immersed surfaces in three-dimensional normed spaces

In this paper we study curvature types of immersed surfaces in three-dimensional (normed or) Minkowski spaces. By endowing the surface with a normal vector field, which is a transversal vector field given by the ambient Birkhoff orthogonality, we get an analogue of the Gauss map. Then we can define concepts of principal, Gaussian, and mean curvatures in terms of the eigenvalues of the differential of this map. Considering planar sections containing the normal field, we also define normal curvatures at each point of the surface, and with respect to each tangent direction. We investigate the relations between these curvature types. Further on we prove that, under an additional hypothesis, a compact, connected surface without boundary whose Minkowski Gaussian curvature is constant must be a Minkowski sphere.

math.DG↗

arXiv2017

Surface immersions in normed spaces from the affine point of view

The aim of this paper is to investigate the differential geometry of immersed surfaces in three-dimensional normed spaces from the viewpoint of affine differential geometry. We endow the surface with a useful Riemannian metric which is closely related to normal curvature, and from this we re-calculate the Minkowski Gaussian and mean curvatures. These curvatures are also re-obtained in terms of ambient affine distance functions, and as a consequence we characterize minimal surfaces as the solutions of a certain differential equation. We also investigate in which cases it is possible that the affine normal and the Birkhoff normal vector fields of an immersion coincide, proving that this only happens when the geometry is Euclidean.

math.DG↗

arXiv2017

Some topics in differential geometry of normed spaces

For a surface immersed in a three-dimensional space endowed with a norm instead of an inner product, one can define analogous concepts of curvature and metric. With these concepts in mind, various questions immediately appear. The aim of this paper is to propose and answer some of those questions. In this framework we prove several characterizations of minimal surfaces, and analogues of some global theorems (e.g., Hadamard-type theorems) are also derived. A result on the curvature of surfaces of constant Minkowski width is also given. Finally, we study the ambient metric induced on the surface, proving an extension of the classical Bonnet theorem.

math.DG↗

arXiv2017

On Legendre curves in normed planes

Legendre curves are smooth plane curves which may have singular points, but still have a well defined smooth normal (and corresponding tangent) vector field. Because of the existence of singular points, the usual curvature concept for regular curves cannot be straightforwardly extended to these curves. However, Fukunaga, and Takahashi defined and studied functions that play the role of curvature functions of a Legendre curve, and whose ratio extend the curvature notion in the usual sense. Going into the same direction, our paper is devoted to the extension of the concept of circular curvature from regular to Legendre curves, but additionally referring not only to the Euclidean plane. For the first time we will extend the concept of Legendre curves to normed planes. Generalizing in such a way the results of the mentioned authors, we define new functions that play the role of circular curvature of Legendre curves, and tackle questions concerning existence, uniqueness, and invariance under isometries for them. Using these functions, we study evolutes, involutes, and pedal curves of Legendre curves for normed planes, and the notion of contact between such curves is correspondingly extended, too. We also provide new ways to calculate the Maslov index of a front in terms of our new curvature functions. It becomes clear that an inner product is not necessary in developing the theory of Legendre curves. More precisely, only a fixed norm and the associated orthogonality (of Birkhoff type) are necessary.

math.DG↗

arXiv2017

Discrete Cycloids from Convex Symmetric Polygons

Cycloids, hipocycloids and epicycloids have an often forgotten common property: they are homothetic to their evolutes. But what if use convex symmetric polygons as unit balls, can we define evolutes and cycloids which are genuinely discrete? Indeed, we can! We define discrete cycloids as eigenvectors of a discrete double evolute transform which can be seen as a linear operator on a vector space we call curvature radius space. We are also able to classify such cycloids according to the eigenvalues of that transform, and show that the number of cusps of each cycloid is well determined by the ordering of those eigenvalues. As an elegant application, we easily establish a version of the four-vertex theorem for closed convex polygons. The whole theory is developed using only linear algebra, and concrete examples are given.

math.DG↗

arXiv2017

Closed Cycloids in a Normed Plane

Given a normed plane $\mathcal{P}$, we call $\mathcal{P}$-cycloids the planar curves which are homothetic to their double $\mathcal{P}$-evolutes. It turns out that the radius of curvature and the support function of a $\mathcal{P}$-cycloid satisfy a differential equation of Sturm-Liouville type. By studying this equation we can describe all closed hypocycloids and epicycloids with a given number of cusps. We can also find an orthonormal basis of ${\mathcal C}^0(S^1)$ with a natural decomposition into symmetric and anti-symmetric functions, which are support functions of symmetric and constant width curves, respectively. As applications, we prove that the iterations of involutes of a closed curve converge to a constant and a generalization of the Sturm-Hurwitz Theorem. We also prove versions of the four vertices theorem for closed curves and six vertices theorem for closed constant width curves.

math.DG↗

arXiv2016

Angles in normed spaces

The concept of angle, angle functions, and the question how to measure angles present old and well-established mathematical topics referring to Euclidean space, and there exist also various extensions to non-Euclidean spaces of different types. In particular, it is very interesting to investigate or to combine (geometric) properties of possible concepts of angle functions and angle measures in finite-dimensional real Banach spaces (= Minkowski spaces). However, going into this direction one will observe that there is no monograph or survey reflecting the complete picture of the existing literature on such concepts in a satisfying manner. We try to close this gap. In this expository paper (containing also new results, and new proofs of known results) the reader will get a comprehensive overview of this field, including also further related aspects. For example, angular bisectors, their applications, and angle types which preserve certain kinds of orthogonality are discussed. The latter aspect yields, of course, an interesting link to the large variety of orthogonality types in such spaces.

math.MG↗

arXiv2016

A new construction of Radon curves and related topics

We present a new construction of Radon curves which only uses convexity methods. In other words, it does not rely on an auxiliary Euclidean background metric (as in the classical works of J. Radon, W. Blaschke, G. Birkhoff, and M. M. Day), and also it does not use typical methods from plane Minkowski Geometry (as proposed by H. Martini and K. J. Swanepoel). We also discuss some properties of normed planes whose unit circle is a Radon curve and give characterizations of Radon curves only in terms of Convex Geometry.

math.MG↗

arXiv2016

Geometric constants for quantifying the difference between orthogonality types

This paper is devoted to introduce new geometric constants that quantify the difference between Roberts orthogonality and Birkhoff orthogonality in normed planes. We start by characterizing Roberts orthogonality in two different ways: via bisectors of two points and using certain linear transformations. Each of these characterizations yields one of those geometric constants that we will study.

math.MG↗

arXiv2016

Geometric properties of a sine function extendable to arbitrary normed planes

In this paper we study a metric generalization of the sine function which can be extended to arbitrary normed planes. We derive its main properties and give also some characterizations of Radon planes. Furthermore, we prove that the existence of an angular measure which is "well-behaving" with respect to the sine is only possible in the Euclidean plane, and we also define some new constants that estimate how non-Radon or non-Euclidean a normed plane can be. Sine preserving self-mappings are studied, and a complete description of the linear ones is given. In the last section we exhibit a version of the Law of Sines for Radon planes.

math.MG↗

arXiv2008

Area distances of Convex Plane Curves and Improper Affine Spheres

The area distance to a convex plane curve is an important concept in computer vision. In this paper we describe a strong link between area distances and improper affine spheres. This link makes possible a better understanding of both theories. The concepts of the theory of affine spheres lead to a new definition of an area distance on the outer part of a convex plane arc. Also, based on the theory of discrete affine spheres, we propose fast algorithms to compute the area distances. On the other hand, area distances provide a good geometrical understanding of improper affine spheres.

math.DG↗