Equilateral sets in ` dp
نویسنده
چکیده
If X is a Minkowski space, i.e. a finite dimensional real normed space, then S ⊂ X is an equilateral set if all pairs of points of S determine the same distance with respect to the norm. Kusner conjectured that e(`p) = d +1 for 1 < p < ∞ and e(`1) = 2d [6]. Using a technique combining linear algebra and approximation theory, we prove that for all 1 < p < ∞, there exists a constant Cp > 0 such that e(`p)≤Cpd.
منابع مشابه
Quantization for uniform distributions on equilateral triangles
We approximate the uniform measure on an equilateral triangle by a measure supported on n points. We find the optimal sets of points (n-means) and corresponding approximation (quantization) error for n ≤ 4, give numerical optimization results for n ≤ 21, and a bound on the quantization error for n → ∞. The equilateral triangle has particularly efficient quantizations due to its connection with ...
متن کاملFixed-Orientation Equilateral Triangle Matching of Point Sets
Given a point set P and a class C of geometric objects, GC(P ) is a geometric graph with vertex set P such that any two vertices p and q are adjacent if and only if there is some C ∈ C containing both p and q but no other points from P . We study G5(P ) graphs where 5 is the class of downward equilateral triangles (ie. equilateral triangles with one of their sides parallel to the x-axis and the...
متن کاملIdentifying flaws in the security of critical sets in latin squares via triangulations
In this paper we answer a question in theoretical cryptography by reducing it to a seemingly unrelated geometrical problem. Drápal (1991) showed that a given partition of an equilateral triangle of side n into smaller, integer-sided equilateral triangles gives rise to, under certain conditions, a latin trade within the latin square based on the addition table for the integers (mod n). We apply ...
متن کامل2 1 A ug 2 00 7 The geometry of Minkowski spaces — a survey . Part I Horst Martini
We survey elementary results in Minkowski spaces (i.e. finite dimensional Banach spaces) that deserve to be collected together, and give simple proofs for some of them. We place special emphasis on planar results. Many of these results have often been rediscovered as lemmas to other results. In Part I we cover the following topics: The triangle inequality and consequences such as the monotonici...
متن کاملNumerical investigation on the effects of six control rods arranged in equilateral triangular configurations on fluid flow and forced convection heat transfer from a circular cylinder
The present work deals with heat transfer characteristics as well as fluid flow patterns in laminar flow regime for a circular cylinder with six control rods arranged in equilateral triangular geometries. The computations have been carried out by a finite volume approach using the overset grid method. The unsteady flow at Re= 200 and Pr= 0.7 and 7.0 was examined. The effect of the control rods ...
متن کامل