  One of the fundamental results in Convex Geometry is Busemann's theorem,
which states that the intersection body of a symmetric convex body is convex.
Thus, it is only natural to ask if there is a quantitative version of
Busemann's theorem, i.e., if the intersection body operation actually improves
convexity. In this paper we concentrate on the symmetric bodies of revolution
to provide several results on the (strict) improvement of convexity under the
intersection body operation. It is shown that the intersection body of a
symmetric convex body of revolution has the same asymptotic behavior near the
equator as the Euclidean ball. We apply this result to show that in
sufficiently high dimension the double intersection body of a symmetric convex
body of revolution is very close to an ellipsoid in the Banach-Mazur distance.
We also prove results on the local convexity at the equator of intersection
bodies in the class of star bodies of revolution.

Metric Geometry