Gruber P. Convex and Discrete Geometry

7 Valuations 125
and only if every symmetric proper convex body in E d is an intersection body. For
more precise information and additional references see the survey of Koldobsky and
König [607] and Koldobsky’s book [606]. It is a pity that the Busemann-Petty problem
has a negative answer. Otherwise it would have made a deep and interesting
result.
Equally intuitive is Shephard’s problem [931]:
Problem 7.3. Let C, D ∈ Cp be two o-symmetric convex bodies such that
(14) v(C|H) ≤ v(D|H)
for each (d − 1)-dimensional linear subspace H of E d , where C|H denotes the
orthogonal projection of C into H. Does it then follow that V (C) ≤ V (D)?
The projection theorem of Alexandrov [11] says the following: Let C, D be centrally
symmetric proper convex bodies such that for each hyperplane H the projections
C|H and D|H have the same area. Then C and D coincide up to a translation.
Considering this result, a positive answer to the above question seems plausible.
Unfortunately, things go wrong as much as they possibly can: There are (even centrally
symmetric) convex bodies C and D such that v(C|H) < v(D|H) for all
(d − 1)-dimensional subspaces H of E d , yet V (C) >V (D). Examples were provided
by Petty [796] and Schneider [897]. Petty and Schneider also proved that the
answer is positive, if the body D is a zonoid. For more information, consult Gardner’s
book [359].
In both cases, the question remains to determine precise additional conditions
under which the problems of Busemann-Petty and Shephard have positive answers.
If C, D are proper, o-symmetric convex bodies, then both (13) and (14) imply
that
V (C) ≤ √ dV(D).
This can be shown by the Corollary 11.2 of John’s ellipsoid theorem 11.2, see
Gardner [359], Theorems 4.2.13 and 8.2.13. While in the case of projections, no
essential improvements are possible, √ d can be replaced by essentially smaller quantities
in the case of sections. It is even possible that the slicing problem which has
been investigated intensively in the local theory of normed spaces has a positive
solution. We state the following version of it:
Problem 7.4. Does there exist an absolute constant c > 0 such that the following
inequality holds: Let C, D ∈ Cp be o-symmetric convex bodies such that
v(C ∩ H) ≤ v(D ∩ H)
for each (d − 1)-dimensional subspace H. Then V (C) ≤ cV(D).
For more information see Gardner [359] and, in the local theory of normed spaces,
Giannopoulos and Milman [374, 375].