Gruber P. Convex and Discrete Geometry

168 Convex Bodies
Clearly, the integral here is at least
Hence
µ({x : δC(x) >ε})e 1 4 ε2
.
µ({x : δC(x) >ε}) ≤ 2e − 1 4 ε2
,
which immediately yields (4). ⊓⊔
9 Symmetrization
A convex body is symmetric with respect to a group of transformations if it is invariant
under each transformation of the group. The group may consist of orthogonal,
affine or projective transformations, or, if the convex body is a polytope, of combinatorial
transformations.
For the rich modern theory of symmetric convex and non-convex polytopes
which can be traced back to antiquity (Platonic solids), we refer to Coxeter [230,
232], Robertson [842], Johnson [550] and McMullen and Schulte [717].
Here, our objective is the following: A more symmetric convex body has in
many cases better geometric or analytic properties. It is thus of interest to consider
symmetrization methods which transform convex bodies into more symmetric ones.
The known symmetrization methods all have the useful property that they decrease,
respectively, increase salient geometric quantities such as the quermassintegrals, in
particular the surface area, diameter, width, inradius and circumradius. More important
for applications is the fact that they increase, respectively, decrease electrostatic
capacity, torsional rigidity and the first principal frequency of membranes.
In this section we first study Steiner symmetrization and use it to prove the isodiametric,
the isoperimetric and the Brunn–Minkowski inequalities. Then Schwarz
symmetrization and rearrangement of functions are investigated. Our applications
concern the isoperimetric inequalities of mathematical physics. In particular, we
consider torsional rigidity of rods and the first principal frequency of membranes.
Finally, we investigate central symmetrization and prove the inequality of Rogers and
Shephard. It has applications in the geometry of numbers and in discrete geometry,
see Sects. 30.1 and 30.3.
For more information on the geometric aspects of symmetrization, see the
books of Hadwiger [468], Leichtweiss [640], Gardner [359], and the surveys of
Lindenstrauss and Milman [660], Sangwine-Yager [878] and Talenti [986]. For pertinent
results in mathematical physics we refer to the books of Pólya and Szegö [811],
Bandle [64] and the survey of Talenti [987].
9.1 Steiner Symmetrization
In the following we define Steiner symmetrization, prove several of its properties
and the sphericity theorem of Gross. Blaschke (or Minkowski) symmetrization is
mentioned.
For more information, see the references mentioned above.