Polynomially integrable bodies

Question context

Let $A_{K,u}(t) = |K \cap \{ tu + u^\perp \}|_{n-1}$ be a parallel section function of $K$, then an infinitely smooth convex body $K$ in $\mathbb{R}^n$ is called polynomially integrable if its parallel section functions $A_{K,u}(t)$ are polynomials of $t$ (on its support) for all $u \in S^{n-1}$.

It is known that the only smooth bodies in odd dimensions are ellipsoids, and in even dimensions, such bodies do not exist.

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