Finding the optimal shadows of a convex polytope

Abstract

Let P be a convex polvtope in Rd. We discuss the problem of placing a light source at infinity so as to minimize or maximize the shadow area of the poly tope. By shadow area we mean the [d -l)-volume of the orthogonal projection of P on a hyper- plane normal to the direction of illumination. Let n be the number of [d-l)-dimensional facets of the poly tope. We exhibit two algo-rithms for finding the optimal placement of the light source. One algorithm uses 0(n4-1) space and time to find the optimal placement. The other uses 0(n ) space to find the optimal placement in 0(nrf~llogn) time. Also, we present an interesting result relating the minimum and maximum shadow arens of P to the radii of the inscribed and circumscribed sphere of a zonotope derived from P.