Illumination Bodies in Projective Geometries

We extend the notion of illumination bodies to Riemannian spaces of constant curvature and to projective Finsler geometries. We prove that the derivative of their volume defines a notion of surface area for convex bodies in these settings, generalizing the affine surface area in Euclidean space. The proof is based on a general result on the derivative of weighted volumes of weighted illumination bodies in Euclidean space. In the appendix, we give some explicit examples for non-Euclidean illumination bodies.