On equidistant polytopes in the Euclidean space

TL;DR

This paper studies equidistant polytopes in Euclidean space, exploring their properties, classifications, and special cases like equidistant polygons, with a focus on types involving inner and outer focal points and their geometric characterizations.

Contribution

It introduces a comprehensive framework for understanding equidistant polytopes, including their properties, graph representations, and specific characterizations of equidistant polygons of type (3,2).

Findings

Equidistant polytopes generalize convexity.

Characterization of equidistant polygons of type (3,2).

Constructive recognition process for certain equidistant polygons.

Abstract

An equidistant polytope is a special equidistant set in the space $\mathbb{R}^n$ all of whose boundary points have equal distances from two finite systems of points. Since one of the finite systems of the given points is required to be in the interior of the convex hull of the other one we can speak about inner and outer focal points of the equidistant polytope. It is of type $(q, p)$, where $q$ is the number of the outer focal points and $p$ is the number of the inner focal points. The equidistancy is the generalization of convexity because a convex polytope can be given as an equidistant polytope of type $(q, 1)$, where $q\geq n+1$. In the paper we present some general results about the basic properties of the equidistant polytopes: convex components, graph representations, connectedness, correspondence to the Voronoi decomposition of the space etc. Especially, we are interested in equidistant polytopes of dimension $2$ (equidistant polygons). Equidistant polygons of type $(3,2)$ will be characterized in terms of a constructive (ruler-and-compass) process to recognize them. In general they are pentagons with exactly two concave angles such that the vertices, where the concave angles appear at, are joined by an inner diagonal related to the adjacent sides of the polygon in a special way via the three reflection theorem for concurrent lines. The last section is devoted to some special arrangements of the focal points to get the concave quadrangles as equidistant polygons of type $(3,2)$.