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Csaba Vincze

Publications and source records attributed to Csaba Vincze.

28 records · Page 2Linked to original sources
arXiv2017

Lazy orbits: an optimization problem on the sphere

Non-transitive subgroups of the orthogonal group play an important role in the non-Euclidean geometry. If $G$ is a closed subgroup in the orthogonal group such that the orbit of a single Euclidean unit vector does not cover the (Euclidean) unit sphere centered at the origin then there always exists a non-Euclidean Minkowski functional such that the elements of $G$ preserve the Minkowskian length of vectors. In other words the Minkowski geometry is an alternative of the Euclidean geometry for the subgroup $G$. It is rich of isometries if $G$ is "close enough" to the orthogonal group or at least to one of its transitive subgroups. The measure of non-transitivity is related to the Hausdorff distances of the orbits under the elements of $G$ to the Euclidean sphere. Its maximum/minimum belongs to the so-called lazy/busy orbits, i.e. they are the solutions of an optimization problem on the Euclidean sphere. The extremal distances allow us to characterize the reducible/irreducible subgroups. We also formulate an upper and a lower bound for the ratio of the extremal distances. As another application of the analytic tools we introduce the rank of a closed non-transitive group $G$. We shall see that if $G$ is of maximal rank then it is finite or reducible. Since the reducible and the finite subgroups form two natural prototypes of non-transitive subgroups, the rank seems to be a fundamental notion in their characterization. Closed, non-transitive groups of rank $n-1$ will be also characterized. Using the general results we classify all their possible types in lower dimensional cases $n=2, 3$ and $4$. Finally we present some applications of the results to the holonomy group of a metric linear connection on a connected Riemannian manifold.

math.MG↗
arXiv2017

On spectral analysis in varieties containing the solutions of inhomogeneous linear functional equations

The aim of the paper is to investigate the solutions of special inhomogeneous linear functional equations by using spectral analysis in a translation invariant closed linear subspace of additive/multiadditive functions containing the restrictions of the solutions to finitely generated fields. The application of spectral analysis in some related varieties is a new and important trend in the theory of functional equations; especially they have successful applications in case of homogeneous linear functional equations. The foundation of the theory can be found in M. Laczkovich and G. Kiss \cite{KL}, see also G. Kiss and A. Varga \cite{KV}. We are going to adopt the main theoretical tools to solve some inhomogeneous problems due to T. Szostok \cite{KKSZ08}, see also \cite{KKSZ} and \cite{KKSZW}. They are motivated by quadrature rules of approximate integration.

math.CV↗
arXiv2017

On spectral synthesis in varieties containing the solutions of inhomogeneous linear functional equations

As a continuation of our previous work \cite{KV2} the aim of the recent paper is to investigate the solutions of special inhomogeneous linear functional equations by using spectral synthesis in translation invariant closed linear subspaces of additive/multiadditive functions containing the restrictions of the solutions to finitely generated fields. The idea is based on the fundamental work of M. Laczkovich and G. Kiss \cite{KL}. Using spectral analysis in some related varieties we can prove the existence of special solutions (automorphisms) of the functional equation but the spectral synthesis allows us to describe the entire space of solutions on a large class of finitely generated fields. It is spanned by the so-called exponential monomials which can be given in terms of automorphisms of $\cc$ and differential operators. We apply the general theory to some inhomogeneous problems motivated by quadrature rules of approximate integration \cite{KKSZ08}, see also \cite{KKSZ} and \cite{KKSZW}.

math.CV↗
arXiv2016

The discrete Pompeiu problem on the plane

We say that a finite subset $E$ of the Euclidean plane $\mathbb{R}^2$ has the discrete Pompeiu property with respect to isometries (similarities), if, whenever $f:\mathbb{R}^2\to \mathbb{C}$ is such that the sum of the values of $f$ on any congruent (similar) copy of $E$ is zero, then $f$ is identically zero. We show that every parallelogram and every quadrangle with rational coordinates has the discrete Pompeiu property w.r.t. isometries. We also present a family of quadrangles depending on a continuous parameter having the same property. We investigate the weighted version of the discrete Pompeiu property as well, and show that every finite linear set with commensurable distances has the weighted discrete Pompeiu property w.r.t. isometries, and every finite set has the weighted discrete Pompeiu property w.r.t. similarities.

math.FA↗
arXiv2016

An observation on Asanov's Unicorn metrics

Finsleroid-Finsler metrics form an important class of singular (y-local) Finslerian metrics. They were introduced by G. S. Asanov in 2006. As a special case Asanov produced examples of Landsberg spaces of dimension at least three that are not of Berwald type. These are called Unicorns [5]. The existence of regular (y - global) Landsberg metrics that are not of Berwald type is an open problem up to this day. In this paper we prove that Asanov's Unicorns belong to the class of generalized Berwald manifolds. More precisely we prove the following theorems: a Finsleroid-Finsler space is a generalized Berwald space if and only if the Finsleroid charge is constant. Especially a Finsleroid-Finsler space is a Landsberg space if and only if it is a generalized Berwald manifold with a semi-symmetric compatible linear connection.

math.DG↗
arXiv2016

Analytic properties and the asymptotic behavior of the area function of a Funk metric

In Minkowski geometry the unit ball is a compact convex body $K$ containing the origin in its interior. The boundary of the body is formed by the unit vectors. We also have a so-called Minkowski functional to measure the length of vectors. By changing the origin in the interior of the body we have a smoothly varying family of Minkowski functionals. This is called the Funk metric. Under some regularity conditions the Minkowski functionals allow us to measure the volume (area) of the indicatrix bodies (hypersurfaces). Some homogenity properties provide the volume and the area to be proportional. The area as the function of the base point varying in the interior of $K$ is strictly convex [25]. This is called the area function of the Funk manifold. If the minimum is attained at the origin then $K$ is said to be balanced. The idea comes from the generalization of Brickell's theorem [6] for Finsler manifolds with balanced indicatrices [25]. As a continuation of [25] we are going to investigate analytic properties and the asymptotic behavior of the area function of a Funk manifold. We prove that the area function is locally analytic and the area can be arbitrary large near to the boundary of $K$. Therefore the minimum always attained at a uniquely determined interior point of $K$. If we apply the result to the indicatrices of a Finsler manifold point by point then the uniquely defined minima of the area functions constitute a vector field. We prove that it is differentiable. Therefore each indicatrix body can be translated in such a way that the translated body is balanced and we always have an associated Finsler manifold with balanced indicatrices. Finsler manifolds having balanced indicatrices represent a class of Finsler spaces such that the so-called Brickell's conjecture holds [6], see also [25].

math.DG↗
arXiv2016

On Asanov's Finsleroid-Finsler metrics as the solutions of a conformal rigidity problem

Finsleroid-Finsler metrics form an important class of singular (y-local) Finsler metrics. They were introduced by G. S. Asanov [2] in 2006. As the special case of the general construction Asanov produced singular (y - local) examples of Landsberg spaces of dimension at least three that are not of Berwald type. The existence of regular (y - global) Landsberg metrics that are not of Berwald type is an open problem up to this day; for a detailed exposition of the so-called unicorn problem in Finsler geometry see D. Bao [3]. In this paper we are going to characterize the Finsleroid-Finsler metrics as the solutions of a conformal rigidity problem. We are looking for (non-Riemannian) Finsler metrics admitting a (non-homothetic) conformal change such that the mixed curvature tensor of the Berwald connection contracted by the derivatives of the logarithmic scale function is invariant. We prove that the solutions of class at least $\mathcal{C}^2$ on the complement of the zero section are conformal to Finsleroid-Finsler metrics.

math.DG↗
arXiv2013

Generalized conic functions of hv-convex planar sets: continuity properties and relations to X-rays

In the paper we investigate the continuity properties of the mapping $Φ$ which sends any non-empty compact connected hv-convex planar set $K$ to the associated generalized conic function $f_K$. The function $f_K$ measures the average taxicab distance of the points in the plane from the focal set $K$ by integration. The main area of the applications is the geometric tomography because $f_K$ involves the coordinate X-rays' information as second order partial derivatives \cite{NV3}. We prove that the Hausdorff-convergence implies the convergence of the conic functions with respect to both the supremum-norm and the $L_1$-norm provided that we restrict the domain to the collection of non-empty compact connected hv-convex planar sets contained in a fixed box (reference set) with parallel sides to the coordinate axes. We also have that $Φ^{-1}$ is upper semi-continuous as a set-valued mapping. The upper semi-continuity establishes an approximating process in the sense that if $f_L$ is close to $f_K$ then $L$ must be close to an element $K'$ such that $f_{K}=f_{K'}$. Therefore $K$ and $K'$ have the same coordinate X-rays almost everywhere. Lower semi-continuity is usually related to the existence of continuous selections. If a set-valued mapping is both upper and lower semi-continuous at a point of its domain it is called continuous. The last section of the paper is devoted to the case of non-empty compact convex planar sets. We show that the class of convex bodies that are determined by their coordinate X-rays coincides with the family of convex bodies $K$ for which $f_K$ is a point of lower semi-continuity for $Φ^{-1}$.

math.MG↗
arXiv2013

Average methods and their applications in Differential Geometry I

In Minkowski geometry the metric features are based on a compact convex body containing the origin in its interior. This body works as a unit ball with its boundary formed by the unit vectors. Using one-homogeneous extension we have a so-called Minkowski functional to measure the lenght of vectors. The half of its square is called the energy function. Under some regularity conditions we can introduce an average Euclidean inner product by integrating the Hessian matrix of the energy function on the Minkowskian unit sphere. Changing the origin in the interior of the body we have a collection of Minkowskian unit balls together with Minkowski functionals depending on the base points. It is a kind of special Finsler manifolds called a Funk space. Using the previous method we can associate a Riemannian metric as the collections of the Euclidean inner products belonging to different base points. We investigate this procedure in case of Finsler manifolds in general. Central objects of the associated Riemannian structure will be expressed in terms of the canonical data of the Finsler space. We take one more step forward by introducing associated Randers spaces by averaging of the vertical derivatives of the Finslerian fundamental function. Such kind of perturbation will have a crucial role when we apply the general results to Funk spaces together with some contributions to Brickell's conjecture on Finsler manifolds with vanishing curvature tensor $\mathbb{Q}$.

math.DG↗