# Á. G. HorváthBudapest University of Technology and Economics · Department of Geometry

Á. G. Horváth

Professor

## About

120

Publications

13,255

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341

Citations

Introduction

Additional affiliations

August 1984 - present

**Budapest University of Technology and Economics, Mathematical Institute**

Position

- Head of Department

Education

September 1979 - June 1984

## Publications

Publications (120)

We regard a smooth, $$d=2$$ d = 2 -dimensional manifold $$\mathcal {M}$$ M and its normal tiling M , the cells of which may have non-smooth or smooth vertices (at the latter, two edges meet at 180 degrees.) We denote the average number (per cell) of non-smooth vertices by $$\bar{v}^{\star }$$ v ¯ ⋆ and we prove that if M is periodic then $$\bar{v}^...

The aim of this note is to investigate the properties of the convex hull and the homothetic convex hull functions of a convex body K in Euclidean n-space, defined as the volume of the union of K and one of its translates, and the volume of K and a translate of a homothetic copy of K, respectively, as functions of the translation vector. In particul...

In hyperbolic geometry there are several concepts to measure the breadth or width of a convex set. In the first part of the paper we collect them and compare their properties. Than we introduce a new concept to measure the width and thickness of a convex body. Correspondingly, we define three classes of bodies, bodies of constant with, bodies of co...

We regard a smooth, $d=2$-dimensional manifold $\mathcal{M}$ and its normal tiling $M$, the cells of which may have non-smooth or smooth vertices (at the latter, two edges meet at 180 degrees.) We denote the average number (per cell) of non-smooth vertices by $\bar v^{\star}$ and we prove that if $M$ is periodic then $v^{\star} \geq 2$ and we show...

In this research-expository paper we recall the basic results of reduction theory of positive definite quadratic forms. Using the result of Ryskov on admissible centerings and the result of Tammela about the determination of a Minkowski-reduced form, we prove that the absolute values of coordinates of a minimum vector in a six-dimensional Minkowski...

The aim of this note is to investigate the properties of the convex-hull and the homothetic convex-hull functions of a convex body $K$ in Euclidean $n$-space, defined as the volume of the union of $K$ and one of its translates, and the volume of $K$ and a translate of a homothetic copy of $K$, respectively, as functions of the translation vector. I...

This paper contains a new concept to measure the width and thickness of a convex body in the hyperbolic plane. We compare the known concepts with the new one and prove some results on bodies of constant width, constant diameter and given thickness.

Lovász proved that the chromatic number of the graph formed by
the principal diagonals of an \(n\)-dimensional strongly self-dual polytope is greater
than or equal to \(n+1\). There is equality if the length of the principal diagonals is
greater than the Euclidean diameter of the monochromatic parts of that coloring
of the unit sphere which is base...

In this paper, we propose a well-justified synthetic building of the projective space. We define the concepts of plane and space of incidence and also the statement of Gallucci as an axiom to our classical projective space. To this purpose, we prove from these axioms, the theorems of Desargues, Pappus, and the fundamental theorem of projectivities,...

The “old-new” concept of a convex-hull function was investigated by several authors in the last seventy years. A recent research on it led to some other volume functions as the covariogram function, the widthness function or the so-called brightness functions, respectively. A very interesting fact that there are many long-standing open problems con...

We investigate an inverse problem referring to roulettes in normed planes, thus generalizing analogous results of Bloom and Whitt on the Euclidean subcase. More precisely, we prove that a given curve can be traced by rolling another curve along a line if two natural conditions are satisfied. Our access involves details from a metric theory of trigo...

The "old-new" concept of convex-hull function was investigated by several authors in the last seventy years. A recent research on it led to some other volume functions as the covariogram function, the widthness function or the so-called brightness functions, respectively. A very interesting fact that there are many long-standing open problems conne...

The new result of this paper is connected with the following problem: consider a supporting hyperplane of a regular simplex and its reflected image at this hyperplane. When will the volume of the convex hull of these two simplices be maximal? We prove that in the case when the dimension is less or equal to 4, the maximal volume attained in the case...

The short history and my results on bisectors in a Hungarian talks.

The new result of this paper connected with the following problem: Consider a supporting hyperplane of a regular simplex and its reflected image at this hyperplane. When will be the volume of the convex hull of these two simplices maximal? We prove that in the case when the dimension is less or equal to 4, the maximal volume achieves in that case w...

The new result of this paper connected with the following problem: Consider a supporting hyperplane of a regular simplex and its re ected image at this hyperplane. When will be the volume of the convex hull of these two simplices maximal? We prove that in the case when the dimension is less or equal to 4, the maximal volume achieves in that case wh...

In this paper metric properties of central quadrics and cones in n-dimensional Euclidean space are discussed. Basic statements on the system of confocal regular quadrics are proved, and some interesting analogues of famous three-dimensional theorems are given for higher dimension, such as the Apollonian theorem on pedal curves. We also investigate...

In this paper we deal with problems concerning the volume of the convex hull of two “connecting” bodies. After a historical background we collect some results, methods and open problems, respectively.

In this paper we discuss Chasles’s construction on ellipsoid to draw the semi-axes from a complete system of conjugate diameters. We prove that there is such situation when the construction is not planar (the needed points cannot be constructed with compasses and ruler) and give some others in which the construction is planar.

In this note we prove that the centers of a closed chain of circles for which every two consecutive members meet in the points of two given circles form a tangent polygon of a conic.

In this paper we propose a well-justified synthetic approach of the projective space. We define the concepts of plane and space of incidence and also the Gallucci's axiom as an axiom of the projective space. To this purpose we prove from our axioms, the theorems of Desargues, Pappus-Pascal and the fundamental theorem of projectivities, respectively...

In this paper a special group of bijective maps of a normed plane (or, more generally, even of a plane with a suitable Jordan curve as unit circle) is introduced which we call the group of general rotations of that plane. It contains the isometry group as a subgroup. The concept of general rotations leads to the notion of flexible motions of the pl...

In this paper we discuss Chasles's construction on ellipsoid to draw the semi-axes from a complete system of conjugate diameters. We prove that there is such situation when the construction is not planar (the needed points cannot be constructed with compasses and ruler) and give some others in which the construction is planar.In this paper we discu...

Mit jelent manapság a ”geometria” szó? Kell-e térgeometriát tanulni a középiskolában? Hogy lehet a térszerkezetére hivatkozva síkbeli ”nehéz” tételeket bizonyítani? A matematikatanár-délután keretében tartott előadás a fenti kérdésekre keres választ.

In this lecture we investigate the Chasles construction of an ellipsoid and the wire construction of Staude, respectively. The common concept to help solve these problems is the focal cones from a point of the space of a confocal system of quadrics.

At the center of this seminar lecture we can find the construction of Chasles on conjugate diameters of an ellipsoid. We introduce the so called elliptic coordinates, too.

Chasles construction determines the axes of an ellipsoid from a complete system of conjugate diameters. We investigate the constructibility of this ”construction”.

A gondolkodó embert mindig is foglalkoztatta az a kérdés, hogy az őt körülvevő világ milyen. Ennek leírása pedig csak geometriai fogalmakkal lehetséges, így nem meglepő ezek igen korai feltűnése. Jóval a görög matematika megjelenése előtt, már az egyiptomi matematikában találunk pontos térgeometriai számításokat, természetesen a ma euklideszi geome...

39. Hajós György Országos Matematika verseny, BME Plenáris előadás a díjkiosztó ünnepségen

We give a non-Paschian plane based on the property of betweenness which cannot be derived from an ordering of the points of a line. In this model there is no possibility to define the congruence of segments but we can define angle, triangle and angle measure, respectively. With respect to our definitions the plane has an elliptic character, meaning...

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 con...

Slides of a talk on the Kerékjártó seminar at Szeged march, 2017

The lack of an inner product structure in Banach spaces yields the motivation
to introduce a semi-inner product with a more general axiom system, one missing
the requirement for symmetry, unlike the one determing a Hilbert space. We use
it on a finite dimensional real Banach space $(\X, \| \cdot\|)$ to define and
investigate three concepts. First,...

In this paper we give lower and upper bounds for the volume growth of a
regular hyperbolic simplex, namely for the ratio of the $n$-dimensional volume
of a regular simplex and the $(n-1)$-dimensional volume of its facets. In
addition to the methods of U. Haagerup and M. Munkholm we use a third volume
form is based on the hyperbolic orthogonal coord...

In this paper we overview the theories of conics and roulettes in four non-Euclidean planes, respectively. We collected the literature connected with these classical concepts from the eighteenth century to the present mentioned papers available only on the ArXiv, too.

In this paper we investigate the problem of finding the maximum volume polytopes, inscribed in the unit sphere of the d-dimensional Euclidean space, with a given number of vertices. We solve this problem for polytopes with d+2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepacka...

The slides of the invitad lecture.

We give the basic formulas of special relativity in a time-space defined by
an earlier paper of the author. We also give the concept of time-space manifold
especially the concept of homogeneous time-space manifold. A homogeneous
time-space manifold is a topological manifold allowed its tangent spaces with
the same fixed time-space. In a homogeneous...

In this paper we deal with the problem to find the maximal volume polyhedra
with a prescribed property and inscribed in the unit sphere. We generalize
those inequality (called by \emph{icosahedron inequality}) of L. Fejes-T\'oth
of which an interesting consequence the fact that regular icosahedron has
maximal volume in the class of the polyhedra wi...

Using the method of C. V\"or\"os, we establish results on hyperbolic plane
geometry, related to triangles. In this note we investigate the orthocenter,
the concept of isogonal conjugate and some further center as of the symmedian
of a triangle. We also investigate the role of the "Euler line" and the
pseudo-centers of a triangle.

This paper contains the short history of our department from the origin to 2015 (in Hungarian).

In this paper we deal with problems concerning the volume of the convex hull
of two "connecting" bodies. After a historical background we collect some
results, methods and open problems, respectively.

Using the method of C. V\"or\"os, we establish several results on hyperbolic
plane geometry, related to triangles and circles. We present a model
independent construction for Malfatti's problem and several (more then fifty)
trigonometric formulas for triangles.

In this note we investigate the behavior of the volume that the convex hull
of two congruent and intersecting simplices in Euclidean $n$-space can have. We
prove some useful equalities and inequalities on this volume. For the regular
simplex we determine the maximal possible volume for the case when the two
simplices are related to each other via r...

In earlier papers we changed the concept of the inner product to a more
general one, to the so-called Minkowski product. This product changes on the
tangent space hence we could investigate a more general structure than a
Riemannian manifold. Particularly interesting such a model when the negative
direct component has dimension one and the model sh...

In this note we examine the volume of the convex hull of two congruent copies
of a convex body in Euclidean $n$-space, under some subsets of the isometry
group of the space. We prove inequalities for this volume if the two bodies are
translates, or reflected copies of each other about a common point or a
hyperplane containing it. In particular, we...

In this paper we review the transformations of a Minkowski $n$-space. We also
prove some new results: we give the normal form of the adjoint abelian or
isometric operators of a finite-dimensional real Banach space and also
characterize the isometry group of a Minkowski $3$-space with property that its
unit sphere does not contain ellipse.

In this paper we propose a method to construct probability measures on the
space of convex bodies with a given pushforward distribution. Concretely we
show that there is a measure on the metric space of centrally symmetric convex
bodies, which pushforward by the thinness mapping produces a probability
measure of truncated normal distribution on the...

This is a book (was written in Hungarian) about absolute geometry, elliptic geometry, congruency, models, analytic geometry, volumes, polytopes and space-time. Not only for mathematicians but everybody whose interested in the thought of the last three century.

More than two centuries ago Malfatti raised and solved the following problem
called by Malfatti's construction problem: In a triangle describe three small
circles, each of them touching the other two, and also two sides of the
triangle. Interesting fact that nobody investigated this problem on the
hyperbolic plane, while the case of the sphere were...

This paper collects some important formulas on hyperbolic volume. To determine concrete values of volume function is a very hard question requiring the knowledge of various methods. Our goal to give a non-elementary integral on the volume of the orthosceme (obtain it without using the Schl\"afli differential formula), using edge-lengthes as the onl...

The tangent hyperplanes of the "manifolds" of this paper equipped a so-called Minkowski product. It is neither symmetric nor bilinear. We give a method to handing such an object as a locally hypersurface of a generalized space-time model and define the main tools of its differential geometry: its fundamental forms, its curvatures and so on. In the...

We study the generalized analogues of conics for normed planes by using the
following natural approach: It is well known that there are different metrical
definitions of conics in the Euclidean plane. We investigate how these
definitions extend to normed planes, and we show that in this more general
framework these different definitions yield, in a...

In this paper we shall investigate the following problem: Which is the largest regular triangle in a brick with volume 1 and having edge lengthes greater than $\frac{1}{\sqrt{2}}$? We prove that there are three optimal cases in all which the edge lengthes of the triangles are $\sqrt{2}$, respectively. Comment: 6 pages, 4 figures

In this paper using the concept of the extended Hamming code we give a construction for dense packing of points at distance at least one in such unit cubes which dimension are a power of two. Comment: 6 pages

In this paper we develop the theories of normed linear spaces and of linear spaces with indefinite metric, for finite dimensions both of which are also called Minkowski spaces in the literature.In the first part of this paper we collect the common properties of the semi- and indefinite inner products and define the semi-indefinite inner product as...

It is well known that the description of topological and geometric properties of bisectors in normed spaces is a non-trivial subject. In this paper we introduce the concept of bounded representation of bisectors in finite dimensional real Banach spaces. This useful notion combines the concepts of bisector and shadow boundary of the unit ball, both...

We discuss the concept of the shadow boundary of a centrally symmetric convex ball $K$ (actually being the unit ball of a Minkowski normed space) with respect to a direction ${\bf x}$ of the Euclidean n-space $R^n$. We introduce the concept of general parameter spheres of $K$ corresponding to this direction and prove that the shadow boundary is a t...

We rewrite the property of confocality with respect to a pseudo-Euclidean space. Our observation is that the generalized definition of confocality in $\cite{stachel}$ does not give back to the original definition of confocality of Euclidean conics. The "confocality property" of $\cite{stachel}$ can be get from our definition in the case when the pr...

Using selfadjoint regular endomorphisms, the authors of Stachel and Wallner (Sib Math J 45(4):785–794, 2004) defined, for an indefinite inner product, a variant of the notion of confocality for the Euclidean space. Our aim is to give a definition that is a common generalization of the usual confocality, and the variant in Stachel and Wallner (Sib M...

This paper presents a result concerning the connection between the parallel projection P
v,H
of a parallelotope P along the direction v (into a transversal hyperplane H) and the extension P + S(v), meaning the Minkowski sum of P and the segment S(v) = {λv | −1 ≤ λ ≤ 1}. A sublattice L
v
of the lattice of translations of P associated to the directio...

We discuss the concept of the bisector of a segment in a Minkowski normed n-space. We prove that all bisectors are topological images of a plane of the embedding Euclidean 3-space iff the shadow boundaries of the unit ball K are topological circles. To a conjectured proving strategy for dimensions n, we introduce the concept of general parameter sp...

We discuss the concept of the bisector of a segment in a Minkowski normed n-space. We prove that all bisectors are topological images of a plane of the embedding Euclidean 3-space i the shadow boundaries of the unit ball K are topological circles. To a conjectured proving strategy for dimensions n, we introduce the concept of general parameter sphe...

In this paper we will show and prove a model-independent construction of the normal transversal of two skew lines in the hyperbolic 3-space.

In this paper we shall give a recursion and a new explicite formula for some functions connected with the weight distribution of the second-order Reed–Muller code. We define some new subcodes of it and determine their information rates, respectively.

In this paper we will show that if the long diagonals of a 2n-polygon with equal angles meet at one point then the common perpendicular of its opposite sides also contains this point. Furthermore the polygon is centrally symmetric and regular if and only if the distance of the sides to this point are equal to each other. We give an analogous statem...

Trends and practices of department of geometry between anniversaries 40 and 50 were discussed. The courses such as models of geometries, space concepts in physics and geometry, computer aided geometric modeling, discrete geometry, differential group actions are tought in 4 semesters in rotation. Department also offers PhD and participates in intern...

In this paper we will show and prove a model-independent construction of the normal transversal of two skew lines in the hyperbolic 3-space.

L. Lov¶asz raised the problem in [1] whether 27congruentbricks of edgelengths a, b, c(0 0), concerning the packing or four rectangles of edge lengths a, b into a squareofedgelength a+b.) Hence, fundamentally, this is a special packing problem: some bricks having flxedvolumemustbeputintoacontainerofgivenvolume. Fromthecombinatorial point of view, si...

We discuss the concept of the bisector of a segment in a Minkowski normed n-space, and prove that if the unit ball K of the space is strictly convex then all bisectors are topological images of a hyperplane of the embedding Euclidean n-space. The converse statement is not true. We give an example in the three-space showing that all bisectors are to...

In this paper we shall investigate the boundary of an extremal body K. Using the characterization of the extremal bodies proved by Venkov and McMullen (Theorem 1) we give two theorems (Theorems 4,5) determining the relative position of the lattice vectors on the boundary of K. These statements are analogues of Theorem 2 and Theorem 3 proved for Dir...

In this paper we will investigate an isoperimetric type problem in lattices. If K is a bounded O-symmetric (centrally symmetric with respect to the origin) convex body in En of volume v(K) = 2n det L which does not contain non-zero lattice points in its interior, we say that K is extremal with respect to the given lattice L. There are two variation...

Horvath, AG 5th Intuitive Geometry Conference - Intuitive Geometry SEP 03-08, 1995 BUDAPEST, HUNGARY

This paper consists of two results concerning the Dirichlet-Voronoi cell of a lattice. The first one is a geometric property of the cell of an integral unimodular lattice while the second one gives a characterization of all those lattice vectors of an arbitrary lattice whose multiples by 1/2 are on the boundary of the cell containing the origin. Th...

This second part of my paper discusses the determination of the DIRICHLET-VORONOI cell of a lattice of dimension n. We give some concrete D-V cells and their automorphism groups, the cells of the lattices An, Dn and En, respectively. These lattices are root lattices to the corresponding finite root systems An, Dn, En. They have an important role in...