Polytope Projection and Projection Polytopes

Abstract

Imagine yourself as the commander of a space ship. Liftoff was a piece of cake, and since then you have been gliding merrily along. But then comes the bad news: A Klingon ship is approaching, and you must prepare for the attack. More bad news: Your batteries are running low! The good news is that your solar cells are working and you are close to a bright star. Thus you can recharge your batteries, but you must certainly do that as quickly as possible. You analyze the situation. Since the solar cells are distributed evenly over the surface of the ship, you decide that you should rotate the ship so that its "face area" is maximized with respect to the light source (assuming that you are still so far from the star that the incoming rays are practically parallel). A similar but opposite problem arises when you approach a star that emits harmful radiation. You then want to minimize the exposure to the radiation and therefore to minimize the face area in the direction of the star. In these problems, you are in control of a body in 3, and you want to turn the body so as to maximize or minimize its "shadow area"with respect to a particular direction of projection (the direction of the incoming rays). In a mathematically equivalent formulation, you may regard the body as being fixed and then look for a direction that maximizes or minimizes the area of the body's projection on a plane orthogonal to the direction. Projections belong to the basic tools in many areas of mathematics. While the projection on a given subspace can be expressed as a simple matrux operation applied to the original body, it is not so clear how to find projections that are "optimal"with respect to an application that one may have in mind. Problems of this kind occur in a great variety of situations with a similarly great variety of (more or less explicit) criteria for what is a good projection. Examples include the analysis of statistical, astronomical or linguistic data, and also the design and analysis of algorithms for manifold applications. We do not want to elaborate on these applications here; the goal of this paper really is to present some of the (as we hope the reader will agree) beautiful mathematics underlying the special projection problems of maximizing or minimizing the "shadow area" and their higherdimensional analogues involving orthogonal projections of a body in 114Z1 onto an (n1)-dimensional subspace. We assume that the body in question is an ndimensional convex polytope. When n = 3, this seems to be a reasonable assumption in the case of the space ship (see Figure 1). It is not hard to see that when n = 2 (so that we are projecting a convex polygon P onto various lines), the maximum projection-length is equal to P's diameter and the minimum projection-length is equal to P's width (the minimum distance between two parallel supporting lines of P) (see Figure 2). Thus the n-dimensional task considered here is one of several ways of extending to 114Z1 the classical Euclidean task of computing the diameter and the width of a polygon.