Half of an antipodal spherical design

We investigate several antipodal spherical designs on which we can choose half of the points, one from each antipodal pair, such that they are balanced at the origin. In particular, root systems of type A, D and E, minimal points of the Leech lattice, and the unique tight 7-design on \(S^{22}\) are studied. We also study a half of an antipodal spherical design from the viewpoint of association schemes and spherical designs of harmonic index T.     

Lösbarkeit Nichtlinearer Gleichungen Mit Orthogonalitätssätzen Vom Borsukschen Typ

In this paper we will prove two theorems which are similar to BORSUKs antipodal theorem on the n-dimensional sphere Sⁿ. However, instead of antipodal pairs of points used in BORSUKs theorem, orthogonal pairs of points are regarded. By these “orthogonal versions” of BORSUKs theorem we get existence theorems about the solutions of non-odd equations F(x) = 0 in Rⁿ and on fixed-point-equations in Hilbert spaces.     

A simple scheme for generating nearly uniform distribution of antipodally symmetric points on the unit sphere

A variant of the Thomson problem, which is about placing a set of points uniformly on the surface of a sphere, is that of generating uniformly distributed points on the sphere that are endowed with antipodal symmetry, i.e., if x is an element of the point set then -x is also an element of that point set. Point sets with antipodal symmetry are of special importance to many scientific and engineering applications. Although this type of point sets may be generated through the minimization of a slightly modified electrostatic potential, the optimization procedure becomes unwieldy when the size of the point set increases beyond a few thousands. Therefore, it is desirable to have a deterministic scheme capable of generating this type of point set with near uniformity. In this work, we will present a simple deterministic scheme to generate nearly uniform point sets with antipodal symmetry.     

Borsuk's theorem through complementary pivoting

In this short note a simple and constructive proof is given for Borsuk's theorem on antipodal points. This is done through a special application of the complementary pivoting algorithm.     

A combinatorial proof of the Borsuk-Ulam antipodal point theorem

We give a proof of Tucker’s Combinatorial Lemma that proves the fundamental nonexistence theorem: There exists no continuous map fromB ⁿ toS ^{n − 1} that maps antipodal points of∂B ⁿ to antipodal points ofS ^{n − 1}.     

Rigidity of spherical buildings and joins

We prove a rigidity and a characterization result for buildings and spherical joins using sets of antipodal points.Received: March 2004 Revised: September 2004 Accepted: October 2004     

On the number of antipodal or strictly antipodal pairs of points in finite subsets ofR d,II

The paper is a continuation of [MM], namely containing several statements related to the concept of antipodal and strictly antipodal pairs of points in a subsetX ofR ^d , which has cardinalityn. The pointsx _i, x_jX are called antipodal if each of them is contained in one of two different parallel supporting hyperplanes of the convex hull ofX. If such hyperplanes contain no further point ofX, thenx _i, x_j are even strictly antipodal. We shall prove some lower bounds on the number of strictly antipodal pairs for givend andn. Furthermore, this concept leads us to a statement on the quotient of the lengths of longest and shortest edges of speciald-simplices, and finally a generalization (concerning strictly antipodal segments) is proved.Research (partially) supported by Hungarian National Foundation for Scientific Research, grant no. 1817     

On a class of partial planes related to biplanes

In this note we consider partial planes in which for each element x (point or line) there exists a unique opposite element or antipode x* which cannot be joined to x or has no intersection with x. We also require the existence of a triangle. Such partial planes will be called antipodal planes. We are mainly interested in the subclass of regular antipodal planes satisfying: p I L implies p* I L* for all points p and lines L. We shall provide a free construction of infinite regular antipodal planes. The objects thus constructed are not free objects in the usual sense since between antipodal planes there do not exist proper homomorphisms. On the other hand, regular antipodal planes do have a canonical homomorphic image which is a biplane (cf. Payne, J Comb Theory A 12:268–282, 1972). Regular antipodal planes can be coordinatized by certain algebraic systems in a similar way as projective planes are coordinatized by ternary rings. Again by a free construction, we shall provide examples satisfying a configuration theorem comparable to the Fano condition with fixed line at infinity.     

On antipodal and adjoint pairs of points for two convex bodies

The numbers of antipodal and of adjoint pairs of points are estimated for a given pair of disjoint convex bodies inE ^d .     

Antipodality in hyperbolic space

An antipodal set in Euclidean n-space is a set of points with the property that through any two of them there is a pair of parallel hyperplanes supporting the set. In this paper we discuss the various possible ways to translate this notion to hyperbolic space and find the maximal cardinality of a hyperbolic antipodal set (according to the different definitions). The first two authors were partially supported by the Hung. Nat. Sci. Found. (OTKA), grant no. T043556 and T037752 and the first author was partially supported also by a Natural Sciences and Engineering Research Council of Canada Discovery Grant.