This second edition incorporates some 60 extra pages of material, including seven new figures, another 21 chapter notes, the new Sections 4.4 and A.4, and about 300 additional references. This expansion indicates the amazingly rapid development of geometric tomography over the decade since the first edition appeared. Despite this, the list of 66 open problems is of roughly the same length.

Many corrections have been made. The most significant amendment appears in Chapter 8, written for the first edition very shortly after the pioneering work was published. Alex Koldobsky's work described in Note 8.9 brought to light an error in the solution of the Busemann–Petty problem. The revised Chapter 8 contains the corrected solution, while the six new notes for Chapter 8 struggle to keep pace with the incredible activity around intersection bodies. The task of surveying recent developments would have been a great deal more difficult but for the publication of Koldobsky's fine book [465]. This takes an almost entirely analytic point of view, whereas in Chapter 8 the original geometrical approach is retained as far as possible.

The new Section 4.4 describes an algorithm constructed by the author and an electrical engineer, Peyman Milanfar. It employs another algorithm designed for the reconstruction of a convex body from its surface area measure, the topic of the new Section A.4. Two reasons lie behind the choice of this material.