On algebraically integrable bodies

In 1687, Sir Isaac Newton  established that the area cut off from an oval in $\mathbb R^2$

by a straight line never depends algebraically on the line (the question was motivated by

Kepler's law in celestial mechanics). In 1987, V. I. Arnold proposed to generalize Newton's

observation to higher dimensions and conjectured that all smooth bodies, with the exception

of ellipsoids in odd-dimensional spaces, have an analogous property. The talk is devoted to

the current status of the conjecture