On a special class of equidistant sets in the Euclidean space

TL;DR

This paper investigates a new class of equidistant sets in Euclidean space, called equidistant functions, focusing on cases where these sets can be represented as graphs of functions, including quadratic approximations of focal sets.

Contribution

It introduces and analyzes a new type of equidistant functions arising from quadratic approximations of focal sets, extending previous linear models.

Findings

Derived necessary and sufficient conditions for the existence of equidistant points.

Characterized upper/lower equidistant functions and their properties.

Explored the relationship between equidistant functions and the minimum operator.

Abstract

An equidistant set in the Euclidean space consists of points having equal distances to both members of a given pair of sets, called focal sets. Since there is no effective formula to compute the distance of a point and a set, it is hard to determine the points of an equidistant set in general. Therefore, it is important to investigate some special cases. In the paper we investigate equidistant sets that can be given as the graph of a function. They are called equidistant functions. In the previously examined conceptual model, one of the focal sets is the horizontal hyperplane through the origin and the other one is the epigraph of a positive-valued, continuous function. The equidistant points form the graph of another function over the hyperplane. In a general situation, the hyperplane is the first-order (linear) approximation for one of the focal sets. A natural idea is to substitute the hyperplane by a circle (sphere) as a second-order (quadratic) approximation for one of the focal sets in more complicated cases. Such a generalization results in a new type of equidistant functions we are going to investigate in the present paper. Before considering the special cases in detail, we present some general observations: a necessary and sufficient condition for the existence of equidistant points along the vertical lines, upper/lower equidistant functions, equidistant functions, a necessary and sufficient condition for the existence of the equidistant function, equidistant functions and the minimum operator (a kind of commuting property).