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.     

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.     

Some inequalities involving medians of two simplexes

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

Triangle inequalities associated with the vertex angles and dihedral angles of a simplex and their applications

In this paper, a class of triangle inequalities associated with the vertex angles and dihedral angles of a simplex is given. As an application, we improve an inequality on the volume of the pedal simplex. Received 3 July 2000.     

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.     

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.     

Analytic and geometric isoperimetric inequalities

We prove uncountably many new analytic and geometric isoperimetric inequalities associated with the solutions of second order ordinary differential equations.     

Proof of Oppenheim's area inequalities for triangles and quadrilaterals

Leta ₁,b ₁,c ₁,A ₁ anda ₂,b ₂,c ₂,A ₂ be the sides and areas of two triangles. Ifa=(a ₁ ^p +a ₂ ^p )^1/p ,b=(b ₁ ^p +b ₂ ^p )^1/p ,c=(c ₁ ^p +c ₂ ^p )^1/p , and 1p4, thena, b, c are the sides of a triangle and its area satisfiesA ^p/2A ₁ ^p/2 +A ₂ ^p/2 . If obtuse triangles are excluded,p>4 is allowed. For convex cyclic quadrilaterals, a similar inequality holds. Also, leta, b, c, A be the sides and area of an acute or right triangle. Iff(x) satisfies certain conditions,f(a),f(b),f(c) are the sides of a triangle having areaA _f, which satisfies (4A _f/3)^1/2f((4A/3)^1/2).     

The minimum number of acute dihedral angles of a simplex

For any -dimensional simplex , we confirm a conjecture of Klamkin and Pook (1988) that there are always at least acute dihedral angles in .

     

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.     

Laguerre planes of even order and translation ovals

Translation Laguerre planes of even order are represented in high dimensional projective space over GF(2) by a collection of subspaces that satisfies a very simple condition.This research was supported for the respective authors by a grant from the University of Canterbury and by a Feodor Lynen Fellowship.     

Mappings preserving area 1 of triangles

This paper is concerned with the characterization of area-preserving mappings of real inner product spaces.     

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.     

On the Kneser–Poulsen conjecture in elliptic space

The Kneser–Poulsen conjecture claims that if some balls of Euclidean space are rearranged in such a way that the distances between their centers do not increase, then neither does the volume of the union of the balls. A special case of the conjecture, when the balls move continuously in such a way that the distances between the centers (weakly) decrease during the motion, is known to hold not only in Euclidean, but also in spherical and hyperbolic spaces. In the present paper, we show that this theorem cannot be extended to elliptic space by constructing three smoothly moving congruent balls with centers getting closer to one another in such a way that the volume of the union of the balls strictly increase during the motion. In spite of this counterexample, it is true that n + 1 balls in n-dimensional elliptic space cover maximal volume if the distances between the centers are all equal to the diameter π/2 of the space. The second part of the paper is devoted to the proof of this fact.

---

The authors were supported by the Hung. Nat. Sci. Found. (OTKA), grant no. T047102 and T037752.     

On monotonicity of the hypersphere volume and area

Motivated by reference [6] and applications in signal processing for communications systems, in this note, we obtain the dimension of maximum volume and the dimension of maximum surface area for the hypersphere of constant radius.      

Homology of dihedral quandles

We solve the conjecture by R. Fenn, C. Rourke and B. Sanderson that the rack homology of dihedral quandles satisfies for p odd prime [T. Ohtsuki, Problems on invariants of knots and 3-manifolds, Geom. Topol. Monogr. 4 (2002) 377-572, Conjecture 5.12]. We also show that contains Z_p for n≥3. Furthermore, we show that the torsion of is annihilated by 3. We also prove that the quandle homology contains Z_p for p odd prime. We conjecture that for n>1 quandle homology satisfies: , where f_n are “delayed” Fibonacci numbers, that is, f_n=f_n−1+f_n−3 and f(1)=f(2)=0,f(3)=1. Our paper is the first step in approaching this conjecture.     

Two-transitive orbits in finite projective planes

The problem of classifying finite projective planes of order n with an automorphism group G and a point orbit on which G acts two-transitively is investigated in considerable detail, under the assumption that has length at last n. Combining old and new results a rather satisfying classification is obtained, even though some cases for orbit lengths n and n + 1 remain unsolved.     

The homeomorphism type of unital-like submanifolds in smooth projective planes

We prove that a compact, connected submanifold of the point space of a smooth projective plane is homeomorphic to a sphere provided that certain intersection properties with lines are satisfied. As an application, we show that the set of absolute points of a smooth polarity in a smooth projective plane of dimension 2l is empty or homeomorphic to a sphere of dimension 2l - 1 or .Received: 13 September 2002