On polynomially integrable convex bodies

Abstract: Let \(K\) be a convex body in \(\mathbb R^n\). The parallel section function of \(K\) in the direction \(\xi\in S^{n-1}\) is defined by \(A_{K,\xi}(t)=\mathrm{vol}_{n-1}(K\cap \{\langle x,\xi\rangle =t\}), \; t\in \mathbb R\). \(K\) is called polynomially integrable (of degree \(N\)) if its parallel section function in every direction is a polynomial of degree \(N\). We prove that the only convex bodies with this property in odd dimensions are ellipsoids. This is in contrast with the case of even dimensions, where such bodies do not exist, as shown by Agranovsky. This is a joint work with A. Koldobsky and A. Merkurjev.