Lightness functions

Variants of the brightness function of a convex body KK in RnRn are investigated. The Lambertian lightness function LK(v,w)LK(v,w) gives the total reflected light resulting from illumination by a light source at infinity in the direction ww that is visible when looking in the direction vv. The partial brightness function RK(v,w)RK(v,w) gives the area of the projection orthogonal to vv of the portion of the surface of KK that is both illuminated by a light source from the direction ww and visible when looking in the direction vv. A class of functions called lightness functions is introduced that includes LKLK and RKRK 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.