On the Isotropicity of Fourier Mean Bodies

In this talk, we study the Fourier transforms of the radial functions of radial mean bodies introduced by Gardner and Zhang in 1998. Motivated by this Fourier-analytic viewpoint, we introduce a new family of star-shaped sets, called Fourier mean bodies. We will focus on the isotropicity of Fourier mean bodies and, in particular, on their geometric behavior when the underlying convex body is in an isotropic position. We will also briefly explain how these bodies are connected to classical objects, including radial mean bodies and intersection bodies.