Equidistant thickness in a space of constant curvature (Q2776898)

This paper deals with the widths (at specified points and in specified directions) of convex bodies in spherical, Euclidean and hyperbolic spaces, and the thicknesses which are their minima. In Euclidean spaces, the width of a set in a specified direction is simply the distance between a pair of parallel supporting lines, and the thickness the minimum of these; all reasonable definitions coincide. In curved spaces, we have no simple way to compare directions at different points; width cannot be measured in a direction alone, but must be measured at a (point, direction) pair. Some artifice is needed to associate a pair of supports, necessarily meeting the body in different locations, to such initial data; and the method chosen will affect the widths, and even thickness, obtained.NEWLINENEWLINENEWLINESome definitions of width (and thickness) found in the literature lead to pathologies in curved spaces. In a spherical space, a set with large diameter may be wide despite every moderate-diameter neighborhood being narrow, due to the width ``at'' a point being determined by supports far away. Conversely, in a hyperbolic space, a convex body with a large insphere may be very narrow if measured at a point far away. (Worse -- the reviewer adds -- a body may be ``thinner'' than a proper subset of itself!). These pathologies are related to the familiar optics of spherical and hyperbolic spaces; remote objects loom in the former and dwindle in the latter.NEWLINENEWLINENEWLINEThe author introduces the equidistant width of a body with respect to a hyperplane splitting that body, based on the perpendicular extent, on each side of the hyperplane, of the body. This turns out to avoid both above-mentioned pathologies. Three (generally nonequivalent) forms of equidistant thickness are proposed, in which the minima are taken over all splitting hyperplanes (the general equidistant thickness), supporting hyperplanes (for which one perpendicular extent is 0, giving the normal equidistant thickness), and median hyperplanes (for which the perpendicular extents are equal, giving the median equidistant thickness). The main result of the paper is a set of three lower bounds (one for each geometry) for the normal equidistant thickness of a simplex; results for the median and general equidistant thicknesses are also given. As the author remarks, some interesting problems remain open.