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<p>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. Having no effective formulas to compute the distance of a point and a set, it is hard to determine the points of an equidistant set in general. Instead of explicit computations we can use computer-assisted methods due to the basic theorem of M. Ponce and P. Santibáñez about the (Hausdorff) convergence of equidistant sets under the convergence of the focal sets. The authors also give an error estimation process in [<CitationRef CitationID="CR2">2</CitationRef>] to approximate the equidistant points. An alternative way of approximation is based on finite focal sets as one of the most important computable classes of equidistant sets [<CitationRef CitationID="CR3">3</CitationRef>], see also [<CitationRef CitationID="CR5">5</CitationRef>]. Special classes of equidistant sets allow us to approximate the equidistant points in more complicated cases. In what follows we have a hyperplane corresponding to the first order (linear) approximation for one of the focal sets and the second one is considered as the epigraph of a function. This idea results in the construction of equidistant functions. In the first part of the paper we prove that the equidistant points having equal distances to the epigraph of a positive-valued continuous function and its domain form the graph of a multivariate function. Therefore such an equidistant set is called a multivariate equidistant function. It is a higher-dimensional generalization of functions in [<CitationRef CitationID="CR4">4</CitationRef>] weakening the requirement of convexity as well. We also prove that the equidistant function one of whose focal sets is constituted by the pointwise minima of finitely many positive-valued continuous functions is given by the pointwise minima of the corresponding equidistant functions. In the second part of the paper we consider equidistant functions such that one of the focal sets is the epigraph of a convex function under some smoothness conditions. Irrespective of the dimension of the space we present a special parameterization for the equidistant points based on the closest point property of the epigraph as a convex set and we give the characterization of the equidistant functions as well. To illustrate how the formulas work we present an example with a hyperboloid of revolution as one of the focal sets.</p>

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On computable classes of equidistant sets: multivariate equidistant functions

  • 期刊论文

  • 发表日期：

    2026年2月19日

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  • Á. Nagy,
  • M. Oláh,
  • M. Stoika,
  • Cs. Vincze

摘要

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. Having no effective formulas to compute the distance of a point and a set, it is hard to determine the points of an equidistant set in general. Instead of explicit computations we can use computer-assisted methods due to the basic theorem of M. Ponce and P. Santibáñez about the (Hausdorff) convergence of equidistant sets under the convergence of the focal sets. The authors also give an error estimation process in [2] to approximate the equidistant points. An alternative way of approximation is based on finite focal sets as one of the most important computable classes of equidistant sets [3], see also [5]. Special classes of equidistant sets allow us to approximate the equidistant points in more complicated cases. In what follows we have a hyperplane corresponding to the first order (linear) approximation for one of the focal sets and the second one is considered as the epigraph of a function. This idea results in the construction of equidistant functions. In the first part of the paper we prove that the equidistant points having equal distances to the epigraph of a positive-valued continuous function and its domain form the graph of a multivariate function. Therefore such an equidistant set is called a multivariate equidistant function. It is a higher-dimensional generalization of functions in [4] weakening the requirement of convexity as well. We also prove that the equidistant function one of whose focal sets is constituted by the pointwise minima of finitely many positive-valued continuous functions is given by the pointwise minima of the corresponding equidistant functions. In the second part of the paper we consider equidistant functions such that one of the focal sets is the epigraph of a convex function under some smoothness conditions. Irrespective of the dimension of the space we present a special parameterization for the equidistant points based on the closest point property of the epigraph as a convex set and we give the characterization of the equidistant functions as well. To illustrate how the formulas work we present an example with a hyperboloid of revolution as one of the focal sets.