In a series of papers, Weil initiated the investigation of translation invariant curvature measures of convex bodies, which include as prime examples Federer's curvature measures. In on-going work, we continue this line of research by introducing new tools to study curvature measures. Our main results suggest that the space of curvature measures, which is graded by degree and parity, is highly structured: We conjecture that each graded component has length at most $2$ as a representation of the general linear group, and we prove this in degree $0$ and $n-2$. Beyond this conjectural picture, our methods yield a characterization of Federer's curvature measures under weaker assumptions.
Based in part on joint work with Jakob Schuhmacher.
