Leírás
The illumination body $\mathcal{I}_\delta(K)$ of a convex body K consists of all points whose convex hull with K adds at most volume $\delta$. In Euclidean space, the volume derivative of $\mathcal{I}_\delta(K)$ as $\delta\to 0^+$ recovers the affine surface area of K.
We extend illumination bodies to Riemannian space forms and projective Finsler geometries via a weighted volume approach. We prove a general limit theorem: the derivative of weighted volume of weighted illumination bodies yields a notion of surface area that coincides with the affine/floating surface area in each geometry, unifying these constructions within a single framework. We also discuss convexity properties, that is, illumination bodies are always convex in the hyperbolic plane, but fail to be convex in spherical geometry.
Joint work with Rotem Assouline and Elisabeth M. Werner.
The talk will also be broadcast on Zoom.