Stability results for sections of convex bodies

It is shown by Makai, Martini, and \'Odor that a convex body $K\subset\mathbb{R}^n$, all of whose maximal sections pass through the origin, must be origin-symmetric. We prove a stability version of this result. We also discuss a theorem of Koldobsky and Shane about determination of convex bodies by fractional derivatives of the parallel section function, and establish the corresponding stability result.