Seminar

 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 $$A_{K,\xi}(t)=\mathrm{vol}_{n-1}(K\cap \{\langle x,\xi\rangle =t\}),\quad t\in \R.$$ $K$ 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 smooth origin-symmetric convex bodies with this property in odd dimensions are ellipsoids, if $N\ge n-1$.  This is in contrast with the case of even dimensions and the case of odd dimensions with $N<n-1$, where such bodies do not exist, as it was recently shown by Agranovsky. Joint work with A. Koldobsky and A. Merkurjev.