  Proximity maps and regions are defined based on the relative allocation of
points from two or more classes in an area of interest and are used to
construct random graphs called proximity catch digraphs (PCDs) which have
applications in various fields. The simplest of such maps is the spherical
proximity map which maps a point from the class of interest to a disk centered
at the same point with radius being the distance to the closest point from the
other class in the region. The spherical proximity map gave rise to class cover
catch digraph (CCCD) which was applied to pattern classification. Furthermore
for uniform data on the real line, the exact and asymptotic distribution of the
domination number of CCCDs were analytically available. In this article, we
determine some appealing properties of the spherical proximity map in compact
intervals on the real line and use these properties as a guideline for defining
new proximity maps in higher dimensions. Delaunay triangulation is used to
partition the region of interest in higher dimensions. Furthermore, we
introduce the auxiliary tools used for the construction of the new proximity
maps, as well as some related concepts that will be used in the investigation
and comparison of them and the resulting graphs. We characterize the geometry
invariance of PCDs for uniform data. We also provide some newly defined
proximity maps in higher dimensions as illustrative examples.
