Convex intersection bodies in three and four dimensions

The paper shows that no origin-symmetric convex polyhedron in R-3 is the in tersection body of a star body. It is shown also that every origin-symmetri c convex body in R-d, for d=3 and 4, can be seen as the intersection body o f a star-shaped set whose radial function satisfies conditions related to s uitable non-integer Sobolev classes.