On a local version of the fifth Busemann-Petty Problem

In  1956, Busemann and Petty  posed a series of questions about
symmetric convex bodies, of which only the first one has been solved.
Their fifth problem asks the following.

Let K be  an origin symmetric convex body in the n-dimensional Euclidean
space and let H_x be a hyperplane passing through the origin orthogonal to
a unit direction x. Consider a hyperplane G parallel to H_x and supporting
to K and let              C(K,x)=vol(K\cap H_x)dist (0, G).
If  there exists a constant C such that for all directions x we have
C(K,x)=C, does it follow that K is an ellipsoid?

We give an affirmative answer to this problem for bodies sufficiently
close to the Euclidean ball in the Banach-Mazur distance. This is a joint
work with Maria Alfonseca, Fedor Nazarov and Vlad Yaskin.