A note on Minkowski planes admitting groups of automorphisms of some large types with respect to homotheties

Monica Klein classified Minkowski planes with respect to linearly transitive subgroups of Minkowski homotheties. She obtained 23 possible types. In this paper we investigate Minkowski planes with respect to groups of automorphism of certain Klein types 12 and higher. We show that types 12 and 14 can only occur in finite miquelian Minkowski planes of order 3 or 5, and we provide examples for such groups. Furthermore, we prove that types 13 and 18 in finite Minkowski planes can only occur in miquelian planes.     

On the Minkowski planes constructed by Artzy and Groh

The Minkowski planes constructed by R. Artzy and H. Groh [1] are characterized among the locally, connected and finite dimensional Minkowski planes as strongly semi-(p, w)-transitive Minkowski planes (see Theorem 2). The types of the Artzy-Groh planes in the typification of the Minkowski planes by M. Klein are determined (see Proposition 4). The second author was supported by a DAAD scholarship for a research visit at TU München. He sincerely thanks the Zentrum Mathematik der TU München for their hospitality.     

Finite Minkowski planes

In this paper we give second characterizations of a certain class of finite Minkowski planes.     

Finite Minkowski planes and embedded inversive planes

We show that each known finite Minkowski plane of order even, contains embedded Miquelian inversive planes, (cf. Proposition 1). Received 2 July 1999.     

Two-transitive Minkowski planes

In this paper we determine all finite Minkowski planes with an automorphism group which satisfies the following transitivity property: any ordered pair of nonparallel points can be mapped onto any other ordered pair of nonparallel points.     

A Family Of 2-Dimensional Minkowski Planes with Small Automorphism Groups

This paper concerns 2-dimensional (topological locally compact connected) Minkowski planes. It uses a construction of J. Jakóbowski [4] of Minkowski planes over half-ordered fields and applies it to the field of reals. This generalizes a construction by A. Schenkel [7] of 2-dimensional Minkowski planes admitting a 3-dimensional kernel. It is shown that most planes in this family of Minkowski planes have 0-dimensional and even trivial automorphism groups.     

Key Distribution Patterns Using Tangent Circle Structures

The problem of key management in a communications network is of primary importance. A key distribution pattern is an incidence structure which provides a secure method of distributing keys in a large network reducing storage requirements. It is of interest to find explicit constructions for key distribution patterns. In O'Keefe [5–7], examples are shown using the finite circle geometries (Minkowski, Laguerre and inversive planes); in Quinn [12], examples are constructed from conics in finite projective and affine planes. In this paper, we construct some examples using the finite tangent-circle structures, introduced in Quattrocchi and Rinaldi [10] and we give a comparison of the storage requirements.     

Flat Circle Planes as Integrals of Flat Affine Planes

In a previous article (Arch. Math. {64} (1995), 75–85) we showed that flat Laguerre planes can be constructed by'integrating' flat affine planes. It turns out that'most' of the known flat Laguerre planes arise in this manner. In this paper we show that similar constructions are also possible in the case of the other two kinds of flat circle planes, that is, the flat Möbius planes and the flat Minkowski planes. In particular, we show that many of the known flat Möbius planes can be constructed by integrating a closed strip taken from a flat affine plane. We also show how flat Minkowski planes arise as integrals of two flat affine planes. All known flat Minkowski planes can be constructed in this manner.     

Nonclassical 4-Dimensional Minkowski Planes Obtained as Brothers of Semiclassical 4-Dimensional Laguerre Planes

We describe the first nonclassical 4-dimensional Minkowski planes and show that they have 6-dimensional automorphism groups. These planes are obtained by a construction of Schroth [18] from generalized quadrangles associated with the semiclassical 4-dimensional Laguerre planes. All 4-dimensional Minkowski planess that can be obtained in this way from the semiclassical 4-dimensional Laguerre planes are determined.     

A remark on the introduction of coordinates in Minkowski planes

Every Minkowski parallel-translation plane in the sense of ARTZY [1] satisfies the quadrangles axiom G of BENZ [4, p. 299]. It follows that the class of all parallel-translation planes coincides with the class of all Minkowski planes over a Tits-nearfield. The results can be extended to Minkowski geometries without the tangency property (called B^*-geometries in [4]).     

Toroidal circle planes that are not Minkowski planes

We construct first examples of circle planes on the torus that are no Minkowski planes, but satisfy the same axiom of joining as flat Minkowski planes. The circle planes constructed by us form a special class ofhyperbola structures (see [4]) or(B*)-Geometrien (see [2]).This research was supported by a Feodor Lynen Fellowship and an ARC International Research Fellowship.     

The ratio of the length of the unit circle to the area of the unit disc in Minkowski planes

In their paper ``An Introduction to Finsler Geometry,' J. C. Alvarez and C. Duran asked if there are other Minkowski planes besides the Euclidean for which the ratio of the Minkowski length of the unit ``circle' to the Holmes-Thompson area of the unit disc equals 2. In this paper we show that this ratio is greater than 2, and that the ratio 2 is achieved only for Minkowski planes that are affine equivalent to the Euclidean plane. In other words, the ratio is 2 only when the unit ``circle' is an ellipse.

     

A classification of Minkowski planes over half-ordered fields

This paper concerns a construction of Minkowski planes over half-ordered fields [5] and [20]. Solving various functional equations the Klein-Kroll types of these Minkowski planes are determined with respect toG- andq-translations and (p, q)-homotheties. Examples for some of the resulting types are given.     

Nearaffine planes

In this paper we develop a theory for nearaffine planes analogous to the theory of ordinary affine translation planes. In a subsequent paper we shall use this theory to give a characterization of a certain class of Minkowski planes.     

A characterization of classical Minkowski planes over a perfect field of characteristic two

We give a new set of axioms defining the concept of (B^*)-plane (i.e. Minkowski plane without the tangency property) and we show that every (B^*)-plane in which a condition similar to the “Fano condition” of Heise and Karzel (see [5, § 3]) holds, is a Minkowski plane over a perfect field of characteristic two. In particular, every finite (B^*)-plane of even order is a Minkowski plane over a field. Consequences for strictly 3-transitive groups are derived from the preceding results; in particular, every strictly 3-transitive set of permutations of odd degree containing the identity is a protective group PGL₂(GF(2 ⁿ )) over a finite field GF(2 ⁿ , for some positive integer n.     

Some Minkowski planes with 3-dimensional automorphism group

A new rather large family of 2-dimensional locally compact topological Minkowski planes with an at least 3-dimensional automorphism group is introduced here. Isomorphism classes and automorphisms of these planes are determined.     

New euclidean theorems by the use of Laguerre transformations — Some geometry of Minkowski (2+1)-space

Examples of the use of Laguerre transformations to discover theorems in the Euclidean and Minkowski planes.     

A New Construction for Minkowski Planes

For any pseudo-ordered field F and some mappings f and g of F into itself we can construct a Minkowski plane such that one derived affine plane is a variation on W. A. Pierce's construction. Moreover, such a Minkowski plane induces nearaffine planes described by H. A. Wilbrink.     

On regular 4-coverings and their application for lattice coverings in normed planes

It is well known that the famous covering problem of Hadwiger is completely solved only in the planar case, i.e.: any planar convex body can be covered by four smaller homothetical copies of itself. Lassak derived the smallest possible ratio of four such homothets (having equal size), using the notion of regular 4-covering. We will continue these investigations, mainly (but not only) referring to centrally symmetric convex plates. This allows to interpret and derive our results in terms of Minkowski geometry (i.e., the geometry of finite dimensional real Banach spaces). As a tool we also use the notion of quasi-perfect and perfect parallelograms of normed planes, which do not differ in the Euclidean plane. Further on, we will use Minkowskian bisectors, different orthogonality types, and further notions from the geometry of normed planes, and we will construct lattice coverings of such planes and study related Voronoi regions and gray areas. Discussing relations to the known bundle theorem, we also extend Miquel’s six-circles theorem from the Euclidean plane to all strictly convex normed planes.     

Analog of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> with a subadditive measure

For a subadditive fuzzy measure (not assumed finite), a Minkowski type triangle inequality, with Choquet integrals in place of Lebesgue integrals, is shown to hold. It is immediate that the set of functions for which a certain positive power of the absolute values have finite Choquet integrals is closed under addition, leading to a linear space analogous to the Lebesgue space L ^p , with a metric related to the integral of that power. Under the additional condition that the subadditive fuzzy measure is inner continuous (Sugeno), the space is shown to be complete. Consequences of the Minkowski type inequality are illustrated in two specific instances.