| Résume | In this talk, we discuss several geometric bodies naturally associated with a convex body, including the classical projection and intersection bodies, as well as radial (p)-mean bodies. These constructions play a central role in geometric tomography and are connected to many intriguing open problems. We then introduce a new construction, the Fourier (p)-mean body of a convex body, and present some of its basic properties, including unique determination, convexity, and affine isoperimetric inequalities. We also discuss connections between Fourier (p)-mean bodies and other classical constructions, as well as their relation to isotropic position. Finally, we show that, for an appropriate range of (p), Fourier (p)-mean bodies are close to ellipsoids in the sense of Hensley’s theorem. This is joint work with Dylan Langharst and Auttawich Manui. |
