Gruber P. Convex and Discrete Geometry

178 Convex Bodies
9.3 Schwarz Symmetrization and Rearrangement of Functions
A relative of Steiner’s symmetrization is Schwarz’s symmetrization. It was introduced
by Schwarz [922] as a tool for the solution of the geometric isoperimetric
problem. While of interest in convex geometry, its real importance is in the context
of isoperimetric inequalities of mathematical physics, where a version of it, the
spherical rearrangement of functions, is an indispensable tool.
In this section Schwarz symmetrization of convex bodies and spherically symmetric
rearrangement of real functions are treated. Without proof we state a result on
rearrangements. Applications are dealt with in the following sections.
For references to the literature on rearrangement of functions and its applications
to isoperimetric inequalities of mathematical physics and partial differential
equations, consult the books of Bandle [64] and Kawohl [569] and the survey of
Talenti [987]. Unfortunately, a modern treatment of this important topic with detailed
proofs, still seems to be missing.
Schwarz Symmetrization
Schwarz symmetrization or Schwarz rounding of a convex body C with respect to a
given line L is defined as follows. For each hyperplane H orthogonal to L and which
meets C, replace C ∩ H by the (d − 1)-dimensional ball in H with centre at H ∩ L
and (d − 1)-dimensional volume equal to that of C ∩ H. The union of all balls thus
obtained is then the Schwarz symmetrization sch C = schL C of C with respect to L.
The version 8.4 of the Brunn–Minkowski theorem implies that sch C ∈ C.
Refining the argument that led to Corollary 9.1, shows that there are hyperplanes
H1, H2,...,all containing L, such that
st Hn ···st H1C → schLC.
As a consequence, many properties of Steiner symmetrization also hold for Schwarz
symmetrization; in particular, the properties listed in Proposition 9.1.
Rearrangement of Functions
It is clear that Schwarz symmetrization extends to non-convex sets (see Fig. 9.3).
Given a Borel function f : E d → R, we apply Schwarz symmetrization to the set
B ={(x, t) : f (x) ≥ t} ⊆E d × R with respect to the line {o}×R: Let
B(t) ={(x, t) : f (x) ≥ t} ⊆E d ×{t} for t ∈ R.
Since f , and thus B, is Borel, each set B(t) is also Borel. If B(t) has infinite measure,
let B r (t) = E d ×{t}; otherwise let B r (t) be the ball in E d ×{t} with centre at (o, t)
and measure equal to that of B(t). Therearrangement f r : E d → R of f then is
defined by
f r (x) = sup{t : (x, t) ∈ B r (t)} for x ∈ E d .