Gruber P. Convex and Discrete Geometry

124 Convex Bodies
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Fig. 7.1. Translative equidissectability of rectangles
Remark. The volume or Jordan measure on C can be extended in two ways to a
valuation on L(C). First, by Groemer’s extension theorem. Second, by the restriction
of the Jordan measure on J which is a valuation on J to L(C) ⊆ J . The uniqueness
part of Groemer’s theorem implies that both extensions coincide.
Comparison of Volumes by Sections and Projections; the Problems of Busemann-
Petty and Shephard and the Slicing Problem
Given two convex bodies C, D, how do their volumes compare? More precisely,
what conditions on sections or projections of C and D guarantee that V (C) ≤ V (D)?
A particularly attractive problem in this context is the following problem of
Busemann-Petty [184], where v(·) is the (d − 1)-dimensional volume.
Problem 7.2. Let C, D ∈ Cp be two o-symmetric convex bodies such that
(13) v(C ∩ H) ≤ v(D ∩ H)
for each (d − 1)-dimensional linear subspace H of E d . Does it follow that V (C) ≤
V (D)?
This problem has attracted a good deal of interest over the last two decades, so we
give the main steps of its solution:
The answer is no for:
d ≥ 12: Larman and Rogers [629]
d ≥ 10: Ball [48]
d ≥ 7: Giannopoulos [373], Bourgain [159]
d ≥ 5: Papadimitrakis [785], Gardner [358], Zhang [1043]
The answer is yes for:
d = 3: Gardner [357]
d = 4: Zhang [1044]
A unified solution for all dimensions using Fourier analysis was given by Gardner,
Koldobsky and Schlumprecht [361]. The answers are based on the notion of intersection
body, introduced by Lutwak [668]. Use is made of the following interesting
result of Lutwak: The solution of the Busemann-Petty problem in E d is positive if
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