标题： 论欧几里得空间中的等距多面体 显示英文标题

标题： On equidistant polytopes in the Euclidean space

摘要： 一个等距多面体是空间中的一个特殊等距集，$\mathbb{R}^n$的所有边界点到两个有限点系统的距离相等。 由于给定点的其中一个有限点系统需要位于另一个点系统的凸包内部，我们可以谈论等距多面体的内焦点和外焦点。 它的类型为$(q, p)$，其中$q$是外焦点的数量，$p$是内焦点的数量。 等距性是凸性的推广，因为一个凸多面体可以表示为类型为$(q, 1)$的等距多面体，其中$q\geq n+1$。 在本文中，我们介绍了关于等距多面体基本性质的一些一般结果：凸分量、图表示、连通性、与空间的Voronoi分解的对应关系等。 特别是，我们感兴趣的是维度为$2$的等距多面体（等距多边形）。 等距多边形类型 $(3,2)$将通过一种构造性（尺规）过程来对其进行表征以识别它们。 一般来说，它们是恰好有两个凹角的五边形，其中凹角出现的顶点通过一个内部对角线以特殊方式与多边形的相邻边相关联，这是通过共线线的三个反射定理实现的。 最后一节专门讨论焦点的一些特殊排列，以得到凹四边形作为等距多边形类型 $(3,2)$。 显示英文摘要

摘要： 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 $(3,2)$.