Lightness functions

Variants of the brightness function of a convex body K in R-n are investigated. The Lambertian lightness function L-K(v, w) gives the total reflected light resulting from illumination by a light source at infinity in the direction w that is visible when looking in the direction v. The partial brightness function R-K(v, w) gives the area of the projection orthogonal to v of the portion of the surface of K that is both illuminated by a light source from the direction w and visible when looking in the direction v. A class of functions called lightness functions is introduced that includes L-K and R-K as special cases. Much of the theory of the brightness function-uniqueness, stability, and the existence and properties of convex bodies of maximal and minimal volume with finitely many function values equal to those of a given convex body-is extended to lightness functions. (C) 2012 Elsevier Inc. All rights reserved.