Gruber P. Convex and Discrete Geometry

280 Convex Polytopes
16 Volume of Polytopes and Hilbert’s Third Problem
By general agreement, a notion of volume of convex polytopes is to be at least simply
additive, translation invariant, non-negative and such that the volume of a cube of
edge-length one equals one. Surprisingly, the proof that there is such a notion seems
to require a limiting argument, for example the exhaustion method. The proof of
the uniqueness is difficult. Rigorous treatments in the context of polytope theory
came forth rather late. We mention Schatunovsky [884], Süss [977] and, in particular,
Hadwiger [468].
A different line of attack is to try to reduce the volume problem for convex polytopes
to that for cubes in the following way, where the volume of a cube is the dth
power of its edge-length. Dissect each polytope into polytopal pieces which, when rearranged
properly, form a cube. The volume of the polytope then is defined to be the
volume of the cube. This is possible for d = 2 but not for d ≥ 3. The latter was shown
by Dehn [250], thereby solving Hilbert’s third problem. While the volume problem
thus cannot be solved by dissections, a rich theory developed around the question
as to when two convex polytopes are G-equidissectable, where G is a group of rigid
motions. Important contributors are, amongst others, Hadwiger, Sydler, Jessen and
Thorup, Sah, Schneider, and McMullen.
In this section, we first show that there is a unique notion of volume for convex
polytopes. Secondly, the equidissectability result for polygons of Bolyai and
Gerwien and the non-equidissectability result of Dehn for regular tetrahedra and
cubes are presented.
For general information the reader is referred to Hadwiger [468], more special
references will be given later.
16.1 Elementary Volume of Convex Polytopes
In Sect. 7.2, we defined the notions of the elementary volume of axis parallel boxes
and of the Jordan measure of convex bodies. Both turned out to be valuations with
special properties. Conversely, it was shown in Sect. 7.3 that valuations with these
properties on the spaces of boxes and convex bodies are, up to multiplicative constants,
the elementary volume and the Jordan measure, respectively.
Here, this program is extended to the elementary volume of convex polytopes,
yet in a strange order. First it will be shown that there is at most one candidate for
the notion of elementary volume. In this part of the proof a simple limiting argument
is needed. Then we present a candidate and show that it has the required properties.
While related to Hadwiger’s [468] proof, the subsequent proof is slightly simpler. An
alternative approach to the elementary volume is mentioned. We also make a remark,
why we don’t simply use Lebesgue or Jordan measure instead of the elementary
volume.
The analogous result for spherical spaces is due to Schneider [903], see also
Böhm and Hertel [135]. A general approach which also treats hyperbolic spaces, was
outlined in McMullen and Schneider [716]. For discussions, see McMullen [714].
We refer also to the simple presentation of Rokhlin [855].