Yaakov S. Kupitz

On the Number of Maximal Regular Simplices Determined by n Points in Rd

A set V = {x 1,…, x n } of n distinct points in Euclidean d-space ℝ d determines 2 n distances ∥x j − x i ∥ (1 ≤ i < j ≤ n). Some of these distances may be equal. Many questions concerning the distribution of these distances have been asked (and, at least partially, answered). E.g., what is the smallest possible number of distinct distances, as a function of d and n? How often can a particular distance (say, one) occur and, in particular, how often can the largest (resp., the smallest) distance occur?

The Fermat-Torricelli point and isosceles tetrahedra

An analytical, unifying approach to geometric properties of the Fermat-Torricelli point of four affinely independent points yields several characterizations of isosceles tetrahedra and, in particular, a characterization or regular tetrahedra within the set of isosceles tetrahedra by means of the solid angle sum.

On the isoperimetric inequalities for Reuleaux polygons

We give a unifying approach to the Blaschke-Lebesgue Theorem and the Firey-Sallee Theorem on Reuleaux polygons in the Euclidean plane.

The Fermat–Torricelli Problem, Part I: A Discrete Gradient-Method Approach

We give a discrete geometric (differential-free) proof of the theorem underlying the solution of the well known Fermat–Torricelli problem, referring to the unique point having minimal distance sum to a given finite set of non-collinear points in d-dimensional space. Further on, we extend this problem to the case that one of the given points is replaced by an affine flat, and we give also a partial result for the case where all given points are replaced by affine flats (of various dimensions), with illustrative applications of these theorems.

Extremal theory for convex matchings in convex geometric graphs

AbstractA convex geometric graphG of ordern consists of the set of vertices of a plane convexn-gonP together with some edges, and/or diagonals ofP as edges. CallG 1-free ifG does not havel disjoint edges in convex position.We answer the following questions:(a)What is the maximum possible number of edges ofG ifG isl-free (as a function ofn andl)?(b)What is the minimum possible number of edges ofG ifG isl-free and saturated, i.e., ifG∪{e} is notl-free for any edge or diagonale ofP that is not, already inG.. We also fully describe the graphsG where the maximum (in (a)) or the minimum (in (b)) is attained. Then we remove the word “disjoint” from the definition of “l-free” and do the same over again. The results obtained are quite similar and closely related to the corresponding results (Turáns theorem, etc) in extremal abstract graph theory.

Separation of a finite set in R d by spanned hyperplanes.

A question of the following kind will concern us here: what is the minimal numbern, ensuring that any spanning set ofn points in 3-space spans a plane, every open side of which contains at least, say, 1000 points of the set. The answer isn=4001 (see Theorem 2.1 below).

Minsum Location Extended to Gauges and to Convex Sets

One of the oldest and richest problems from continuous location science is the famous Fermat–Torricelli problem, asking for the unique point in Euclidean space that has minimal distance sum to

From Intersectors to Successors

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