Showing papers in "Journal of Geometry in 1980"
16 citations
TL;DR: In this article, it was shown that an m-transversal of a family of convex sets in Euclidean n-dimensional space Rn is a m-dimensional flat which intersects each member of the family.
Abstract: An m-transversal of a family of convex sets in Euclidean n-dimensional space Rn is an m-dimensional flat which intersects each member of the family. This paper establishes some results dealing with (n−1)-transversals in Rn. The results are related to a theorem of Hadwiger on 1-transversals in the plane.
15 citations
12 citations
TL;DR: In this paper, the affine Barbilian planes over an arbitrary ring with 1 were characterized and the kernel of the translation Barbilian plane was shown to generalize Leissner's parallelodromic planes.
Abstract: W.Leissner has characterized, by geometric axioms, the affine Barbilian planes over a Z-ring (i.e, a ring with 1 such that ab=1 ⇒ ba=1) [10].The aim of the present paper is to characterize correspondingly the affine Barbilian planes over an arbitrary ring with 1. First we shall deal with the translation Barbilian planes, which generalize Leissner's parallelodromic planes [11]. The paper concludes with a study of the kernel of the translation Barbilian plane.
12 citations
TL;DR: In this article, a noneuclidean regular metric vector space (V,K,Q) is defined and a bijection bijection (V → V) is introduced.
Abstract: Let (V,K,Q) be a noneuclidean regular metric vector space, ρ a fixed element of K and ϕ: V → V a bijection such that
$$Q(x - y) = \rho \leftrightarrow Q(x^\phi - y^\phi ) = \rho \forall x,y \in v.$$
.
9 citations
TL;DR: In this article, the n2-sets in a projective plane were characterized and the exact n2+n lines were completely characterized in terms of n 2 + n lines.
Abstract: In this paper n2-sets, in a projective plane, determining exactly n2+n lines are completely characterized.
8 citations
TL;DR: In this paper, the n2-sets of type (0, 1, n) in projective planes of order greater than 3 were completely characterized, where n 2 is the number of sets of type n.
Abstract: In this paper n2-sets of type (0,1,n), in projective planes of order greater than 3, are completely characterized.
7 citations
TL;DR: In this article, the affine images of convex bodies of constant width (here called affine Gleichdicke) are characterized by having projections which are bounded by P-curves (the non-symmetrical analogues of R-Curves).
Abstract: Characterizing ellipsoids by plane shadow boundaries BLASCHKE also showed that ellipsoids are characterized by the fact that their plane projections are bounded by Radon curves (R-curves). It follows that the affine images of convex bodies of constant width (here called “affine Gleichdicke”) are characterized by having projections which are bounded by P-curves (the non-symmetrical analogues of R-curves). We thereby obtain two further characteristic properties of “affine Gleichdicke”:
5 citations
TL;DR: In this paper, the authors give some new constructions of projective Hjelmslev planes with invariant pairs (t,3) by making use of the generalization and improvement of the Drake-Lenz theorem.
Abstract: Associated with every finite projective Hjelmslev plane is an invariant pair (t,r): t is the number of neighbours of a given point on a given line passing through it and r is the order of the underlying projective plane. The Drake-Lenz method [2],[3] of using auxiliary matrices for the constructions of projective Hjelmslev planes has become standard by now. This paper is intended to give some new constructions of projective Hjelmslev planes with invariant pairs (t,3) by making use of the generalization and improvement of the Drake-Lenz theorem [3] obtained by “the author in [6] and [7]. The results of this paper add 8 new values to the list ([5], example 3.7(ii)) of invariant pairs (t,3) with t ≲ 1,000 for projective Hjelmslev planes.
5 citations
TL;DR: In this article, the uniqueness of Barbilian domains over several classes of commutative rings has been studied, and the authors generalize the results of W.W. Benz and show how to obtain a Barbilian domain from a set of unimodular pairs from R×R.
Abstract: W. Leissner has developed a plane geometry over any Z-ring R, in which a point is an element of R×R and a line is a set of the form {(x+ ra, y + rb):r ∈ R} where (x,y) ∈ R×R and (a,b) is from a “Barbilian domain”, i.e., a set of unimodular pairs from R×R satisfying certain axioms. In this note we generalize results of W. Benz guaranteeing the uniqueness of Barbilian domains over several classes of commutative rings. The author wishes to thank Gordon Keller and Douglas Costa for fruitful discussions, the referee for his improvements, and the University of Virginia for its hospitality while this work was done.
5 citations
TL;DR: In this paper, the authors determine all sets Q of points of any finite dimensional protective space P such that each line intersecting Q in more than one point either is contained in Q or contains exactly one point not on Q. If P is a finite protective space of order q, these sets are the so called sets of class (0, 1, q, q + 1).
Abstract: We determine all sets Q of points of any finite dimensional protective space P such that each line intersecting Q in more than one point, either is contained in Q or contains exactly one point not on Q. If P is a finite protective space of order q, these sets are the so called sets of class (0, 1, q, q + 1).
TL;DR: In this paper, the authors characterize the class of Banach spaces with unique metric lines and complete metric spaces with a unique line in the complete metric space with unique lines, and show that these spaces can be classified into two classes.
Abstract: This paper is concerned with characterizing the class of Banach Spaces with unique metric lines in the class of complete metric spaces with unique lines.
TL;DR: In this paper, it was shown that if γ is a sharply 4 - transitive set of permutations onn elements (n≥7, integer), such that γ contains the identical permutation and γeγ implies γ−1eγ, then isn=11.
Abstract: Under a special assumption, a theorem concerning the order of Minkowski - 2 - Structures is proved. In particular it is shown that if γ is a sharply 4 - transitive set of permutations onn elements (n≥7, integer), such that γ contains the identical permutation and γeγ implies γ−1eγ, then isn=11.
TL;DR: In this paper, extended reflection groups of a metric vector space (V,f) are introduced and defined by a system of generators and a set of defining relations, and it can be proved that they are isomorphic to certain subgroups of the orthogonal groups.
Abstract: The “extended reflection groups” of a metric vector space (V,f) are introduced and defined by a system of generators and a set of defining relations. It can be proved that they are isomorphic to certain subgroups of the orthogonal groups. The main result of the underlying paper is that these groups can be characterized by a few properties among which we mention the validity of the transitivity theorem and the property of “ Δ — intersecting ”. Finally, we obtain a characterization of the (full) groups O*(V,f) in the case dim V⊥≤1.
TL;DR: In this paper, the authors studied three kinds of congruences (pseudospherical, W-and V-congruences) in an isotropic space of degree oneJ3(1), i.e. a three dimensional real affine space with the metricds2=dx2+dy2.
Abstract: In this paper we study three kinds of congruences (pseudospherical, W — andV — congruences) in an isotropic space of degree oneJ3(1), i.e. a three dimensional real affine space with the metricds2=dx2+dy2. It is proved that many results from the euclidean theory are applicable inJ3(1).
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TL;DR: In this article, the affine reflections with axes in a parallel pencil were shown to belong to the commutator subgroup of affine groups in an equiaffine plane.
Abstract: In an equiaffine plane\(\mathfrak{A}\) of characteristic ≠ 2, let\(\mathbb{E}*(\Pi )\) denote the group generated by the affine reflections with axes in a parallel pencil Π. The centre of\(\mathbb{E}*(\Pi )\) is the group of the translations in the direction Π, and the quotient group is isomorphic to the group generated by the affine reflections in the direction Π. Let ρ ∈*\(\mathbb{E}*(\Pi )\). Then ρ is the product of at most two affine reflections unless ρ is a glide reflection in which case three affine reflections are needed. In general, the affine reflections may be required to belong to\(\mathbb{E}*(\Pi )\). We also show that the equiaffine group of\(\mathfrak{A}\) is the commutator subgroup of the affine group.
TL;DR: In this article, it was shown that part of B. Segre's proof is not correct, and that an elliptic quadric ofAG(2,q),q odd, has pseudoregular points iffq=3, 5 or 7.
Abstract: In [1] S. ILKKA conjectured that pqeudoregular points of an elliptic quadric ofAG(2,q),q odd, only exist for small values ofq. In [3] B. SEGRE ”proves” that an elliptic quadric ofAG(2,q),q odd, has pseudoregular points iffq=3 or 5. In [2], however, F. KaRTESZI shows that an elliptic quadric ofAG(2,7) has eight pseudoregular points. In this note we prove that part of B. Segre's proof is not correct, and that an elliptic quadric ofAG(2,q),q odd, has pseudoregular points iffq=3, 5 or 7.
TL;DR: In this article, it was shown that being disjoint with a given block induces an equivalence relation on the block set of such a design, and that any two disjooint blocks meet precisely the same point classes.
Abstract: We consider square divisible designs with parameters n, m, k=r, 0 and λ. We show that “being disjoint” induces an equivalence relation on the block set of such a design and that any two disjoint blocks meet precisely the same point classes. Also, the intersection number of two blocks depends only on their equivalence classes. The number of blocks disjoint with a given block is at most n−1; equality holds for all blocks iff the dual of the given design is also divisible with the same parameters. We then give a few applications.
TL;DR: In this paper, the geometric closure operators on P and P′ are determined whose restriction to P′ equals a given closure operator on P′, such that P′=P−e for some maximal element e e P.
Abstract: If P and P′ are finite partially ordered sets such that P′=P−e for some maximal element e e P, all geometric closure operators on P are determined whose restriction to P′ equals a given closure operator on P′.
TL;DR: In this article, it was shown that π is a pappian plane if (G,S) is an S -group, where S is the collineation group generated by the set S of all these involutory homologies.
Abstract: Let π be a projective plane with the flag (A,b) Assume that there exist all involutory homologies with centres on b and axis through A Let G be the collineation group generated by the set S of all these involutory homologies It is shown that π is a pappian plane if (G,S) is an S - group
TL;DR: In this article, it was shown that generalized euclidean planes (or metric affine planes) can be characterized by a similarity relation on the set of all non-collinear point triples.
Abstract: We show that generalized euclidean planes (or metric affine planes) can be characterized by a similarity relation on the set of all non-collinear point triples or of the set of all point triples of an affine plane. In the second case most of the properties of the relation can be formulated independent of the geometrical structure.
TL;DR: In this article, the limit group (Grenzgruppe) of affine metric n-space (Jn,g) over a field F of char is considered.
Abstract: According to Strubecker an affine metric n-space (Jn,g) over a field F of char. ≠ 2 (n≥2) is called simply-isotropic if Radg is one-dimensional; let Fu be the totally isotropic direction w.r. to g. The group Bg of motions of (Jn,g) contains an invariant (2n−1)-dimensional subgroup G, called thelimit group (Grenzgruppe) which maps planes parallel to Fu into parallel ones. In case n=3, n=5, F=ℝ Strubecker [12], [13], [14] gave several factorizations G=GL oGR of G into a commutative product of subgroups acting 1-transitively (=regularly) on Jn.
TL;DR: In this paper, the authors construct geometrical structures (X,ℒ,//) where X is a finite set of points, ℒ is a set of lines, and // is an equivalence relation on the lines.
Abstract: In this paper,suggested by Andre's papers ([2), [3]), we construct geometrical structures (X,ℒ,//}) where X is a finite set of points, ℒ is a set of lines, and // is an equivalence relation on ℒ. These constructions are made starting with a finite and not empty set X and a permutation group G which is 2-transitive on X and such that the stabilizer of two distinct points of X is different from the identical subgroup. We look for conditions such that the structure (X, ℒ) is a (3,q)-Steiner system. We remember that a (3,q)-Steiner system is a pair (X,B), where X is a set of elements (called points), B is a system of subsets of X (called blocks), such that:
TL;DR: In this paper, the authors study two classes of surfaces in euclidean 3-space, namely, ruled and molding surfaces, special surfaces of revolution (surfaces of revolution are covered by a plane curve if the plane is rolling over a torse).
Abstract: We study two classes of surfaces in euclidean 3-space, namelyruled andmolding surfaces, specialsurfaces of revolution (molding surfaces are covered by a plane curve if the plane is rolling over a torse, in particularsurfaces moulures by G.MONGE for a cylindrical torse). The main result: A connected surface hyperosculating molding surfaces in every point is contained in a ruled or in a molding surface; a connected surface hyperosculating in every point surfaces of revolution is a surface of revolution. We characterize hyperosculating molding surfaces by means of the generating torse and study finally molding surfaces having contact of higher order.
TL;DR: In this article, the authors considered the problem of ordering a rectangular plane to be algebraically represented by a metric integral system and gave necessary and sufficient conditions on the integral system to order the corresponding rectangular plane.
Abstract: H. Karzel [6] has shown that rectangular planes can be algebraically represented by metric integral systems. Here, rectangular planes, which are simultaneously ordered planes, are considered. Necessary and sufficient conditions on the integral system are given for the corresponding rectangular plane to be ordered.
TL;DR: In this article, the authors prove two inequalities for n-dimensional simplexes that can be related to the basic inequalities (1 + 1/n)n < e < (1+ 1/ n)n+1.
Abstract: An n-dimensional form of Winternitz's Theorem for convex sets in En can be related to a standard inequality for e. The object of this note is to prove two inequalities for n-dimensional simplexes that can be related to the basic inequalities (1+1/n)n < e < (1+1/n)n+1.
TL;DR: In this article, the dimension of the Frow spaces of the incidence matrices of all affine planes AG(2,n) is determined, and a unified method for the cases when charF is or is not a divisor of n is presented.
Abstract: The dimension of the F-row spaces of the incidence matrices of all affine planes AG(2,n) is determined. The proof presented here gives a unified method for the cases when charF is or is not a divisor of n.
TL;DR: In this paper, the existence of k-arcs in finite planes of order q with a lower bound b = b(q) for k is proved, where q is the dimension of the plane.
Abstract: We prove the existence of k-arcs in finite planes of order q with a lower bound b=b(q) for k.
TL;DR: In this paper, it was shown that locally Desarguesian spaces represent a generalization other than the obvious one of (certain) Riemannian symmetric spaces, and applications are then given to spaces which are locally symmetric in a wider sense.
Abstract: In Riemannian spaces, locally Desarguesian spaces have constant curvature and are therefore locally symmetric. This does not hold for Finsler spaces, so that locally Desarguesian spaces represent a generalization other than the obvious one we studied previously of (certain) Riemannian symmetric spaces. In this paper we discuss them in detail; as an example of the results obtained we mention that a simply connected locally Desarguesian space without conjugate points is globally Desarguesian. Applications are then given to spaces which are locally symmetric in a wider sense. We also study (and in Minkowski spaces determine exactly) the properties of functions which measure the distance of a point from those on a line.
TL;DR: The finite groups generated by 3-transpositions, studied by B.Fischer [5] have a geometrical interpretation given by F.Buekenhout [1] under the name of Fischer spaces.
Abstract: The finite groups generated by 3-transpositions, studied by B.Fischer [5]have a geometrical interpretation given by F.Buekenhout [1] under the name of Fischer spaces.