Planar sections of the quadric of Lie cycles and their Euclidean interpretations

All cycles (points, oriented circles, and oriented lines of a Euclidean plane) are represented by points of a three dimensional quadric in four dimensional real projective space. The intersection of this quadric with primes and planes are, respectively, two- and one-dimensional systems of cycles. This paper is a careful examination of the interpretation, in terms of systems of cycles in the Euclidean plane, of fundamental incidence configurations involving this quadric in projective space. These interpretations yield new and striking theorems of Euclidean geometry.     

A remark on the faces of the cone of Euclidean distance matrices

A new characterization of the faces of the cone of Euclidean distance matrices (EDMs) was recently obtained by Tarazaga in terms of LGS(D), a special subspace associated with each EDM D. In this note we show that LGS(D) is nothing but the Gale subspace associated with EDMs.     

An Upper Bound for the Cardinality of an <Emphasis Type="Italic">s</Emphasis>-Distance Set in Euclidean Space

In this paper we show that if X is an s-distance set in ^m and X is on p concentric spheres then Moreover if X is antipodal, then .     

Relation between metric spaces and Finsler spaces

In a connected Finsler space Fn=(M,F) every ordered pair of points p,q∈M determines a distance ?F(p,q) as the infimum of the arc length of curves joining p to q. (M,?F) is a metric space if Fn is absolutely homogeneous, and it is quasi-metric space (i.e. the symmetry: ?F(p,q)=?F(q,p) fails) if Fn is positively homogeneous only. It is known the Busemann-Mayer relation , for any differentiable curve p(t) in an Fn. This establishes a 1:1 relation between Finsler spaces Fn=(M,F) and (quasi-) metric spaces (M,?F).We show that a distance function ?(p,q) (with the differentiability property of ?F) needs not to be a ?F. This means that the family {(M,?)} is wider than {(M,?F)}. We give a necessary and sufficient condition in two versions for a ? to be a ?F, i.e. for a (quasi-) metric space (M,?) to be equivalent (with respect to the distance) to a Finsler space (M,F).     

Blockings-dimensional subspaces by lines inPG(2s,q)

We investigate sets of lines inPG(2s,q) such that everys-dimensional subspace contains a line of this set. We determine the minimum number of lines in such a set and show that there is only one type of such a set with this minimum number of lines.     

Birational maps between generalized Severi-Brauer varieties

A conjecture of Amitsur states that two Severi-Brauer varieties V(A) and V(B) are birationally isomorphic if and only if the underlying algebras A and B are the same degree and generate the same cyclic subgroup of the Brauer group. We examine the question of finding birational isomorphisms between generalized Severi-Brauer varieties. As a first step, we exhibit a birational isomorphism between the generalized Severi-Brauer variety of an algebra and its opposite. We also extend a theorem of P. Roquette to generalized Severi-Brauer varieties and use this to show that one may often reduce the problem of finding birational isomorphisms to the case where each of the separable subfields of the corresponding algebras are maximal, and therefore to the case where the algebras have prime power degree. We observe that this fact allows us to verify Amitsur’s conjecture for many particular cases.     

An inequality for a simplex and its applications

In this paper, we improve some inequalities for ann-dimensional simplex in Euclidean spaceE ⁿ and give some applications.     

A NOTE ON CITY BLOCK DISTANCE

In this paper, problem of characterizing the city block distance between two lattice points in k-dimensional Euclidean space is discussed.     

An extension of the Erdős-Szekeres theorem on large angles

The existence of a functionn(ε) (ε>0) is established such that given a finite setV in the plane there exists a subsetW⊆V, |W|<n(ε) with the property that for anyv εV\ W there are two pointsw ₁,w ₂ εW such that the angle ∢(w ₁ vw ₂)>π-ε.     

Generalization and sharpening of Safta's Conjeture in the n-dimensional space

In this paper, a generalization and an improvement of Safta's conjecture in the n-dimensional space are given.     

Relationship between tetrahedron shape measures

Tetrahedron shape measures are used for evaluating the quality of tetrahedra in finite element meshes. Three shape measures, theminimum solid angle _min theradius ratio , and themean ratio , are discussed in this paper. A new formula for the computation of a solid angle of tetrahedron is derived. For different shape measures andv (with values 1), we establish a relationship between andv of the form wherec ₀,c ₁,e ₀, ande ₁ are positive constants. This means that if one measure approaches zero for a poorly-shaped tetrahedron, so does the other. Combined with the property that each measure attains a maximum value only for the regular tetrahedron, this means that the shape measures are equivalent.This work was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada.     

Some inequalities on areas of bisection planes of dihedral angles of a simplex

In this paper we prove some inequalities on areas of bisection planes of dihedral angles of a simplex in E ⁿ.     

A geometric inequality for a finite set of points on a sphere and its applications

In this paper we present a geometric inequality for a finite number of points on an (n–1)-dimensional sphere S _n–1(R). As an application, we obtain a geometric inequality for finitely many points in the n-dimensional Euclidean space E ⁿ.     

Some inequalities involving medians of two simplexes

In this paper, some inequalities involving medians of two simplexes are established.     

On the isoperimetric deficit of a simplex and of a polygon

This paper gives inequalities which estimate the isoperimetric deficit of a simplex and of a polygon and some consequences thereof.     

Three geometric inequalities for a simplex

In this paper, we obtain three new geometric inequalities for ann-dimensional simplex in then-dimensional Euclidean spaceE ⁿ . As special cases we find two known inequalities from L. Fejes Tóth and M. S. Klamkin, respectively.     

Ptolemaic inequalities

Any odd-sided cyclic polygon has a family of alternating inequalities which generalize Ptolemy's theorem; the expressions in the inequalities are weighted sums of the distances from the vertices to a general point. When the polygon is regular there are similar inequalities in higher odd powers of the distances.     

Generalization of the triangle and Ptolemy inequalities

The Euclidean triangle inequality generalizes to an alternating inequality for any oddsided polygon that can be inscribed in a circle; there is equality in the even cases. A generalization of Ptolemy's theorem follows by inversion. The results have Minkowskian analogues.     

Feuerbach's relation and Ptolemy's theorem in R n

Ptolemy's equality for four points on a circle is related to a Feuerbach-type area relation. This suggested an extension of Ptolemy's inequality to a Feuerbach type volume relation between simplexes formed from n+2 points in R ⁿ (n2). Extensions of the Möbius-Neuberg and Pompeiu Theorems in R ² are given for R ⁿ . Ptolemy's inequality is also extended to convex n-gons in R ² yielding an extension of Fuhrmann's hexagon theorem to 2n-gons in R ² (n3).