Gruber P. Convex and Discrete Geometry

9 Symmetrization 171
(iv) Trivial.
(v) Let C, C1, C2, ··· ∈Cp be such that C1, C2, ···→C. We have to show that
st C1, st C2, ···→st C. We clearly may suppose that o ∈ int C. The assumption that
C1, C2, ··· → C is then equivalent to the following: Let ε>0, then (1 − ε)C ⊆
Cn ⊆ (1+ε)C for all sufficiently large n. By (ii) and (iv), this implies the following:
Let ε>0, then (1−ε) st C ⊆ st Cn ⊆ (1+ε) st C for all sufficiently large n. Since
o ∈ int st C, this implies that st C1, st C2, ···→st C.
(vi) This property is an immediate consequence of the definition of the Steiner
symmetrization, (i) and Fubini’s theorem.
(vii) By (iii) and (ii), st(C + εB d ) ⊇ st C + ε st B d = st C + εB d for ε>0.
Hence
V (st C + εB d ) − V (st C)
ε
≤ V � st (C + εB d ) � − V (st C)
ε
= V (C + εBd ) − V (C)
for ε>0,
ε
by (iv) and (vi). Now let ε →+0 and note the definition of surface area in Sect. 6.4.
(viii) Let x, y ∈ st C and let T and st T be as in the proof of (i). Then, at least one
of the diagonals of the trapezoid T ⊆ C has length greater than or equal to �x − y�.
(ix) Let c + ϱB d ⊆ C. Then st (c + ϱB d ) ⊆ st C by (iv) and thus b + ϱB d ⊆ st C
for suitable b. Hence r(C) ≤ r(st C). The inequality R(st C) ≤ R(C) is shown
similarly, using balls containing C. ⊓⊔
Remark. Proposition 9.1(iii) can be refined as follows: Let C, D ∈ Cp. Then
st (C + D) ⊇ st C + st D,
where equality holds if and only if C and D are homothetic. Propositions (vi) and
(vii) admit the following generalization and refinement: Let C ∈ C, then
Wi(st C) ≤ Wi(C) for i = 0, 1,...,d,
where for C ∈ Cp and i = 1,...,d − 1 equality holds if and only if C is symmetric
in a hyperplane parallel to H. See the references at the beginning of Sect. 9.
Polar Bodies and Steiner Symmetrization
Let C be a convex body with o ∈ int C. Itspolar body C ∗ (with respect to o) is
defined by
C ∗ ={y : x · y ≤ 1forx ∈ C}.
It is easy to see that C ∗ is also a convex body with o ∈ int C ∗ . Polar bodies play
a role in geometric inequalities, the local theory of normed spaces, the geometry
of numbers and other areas. The following property was noted by Ball [47] and
Meyer and Pajor [720]. It can be used to prove the Blaschke–Santaló inequality, see
Theorem 9.5. Let v(·) denote (d − 1)-dimensional volume.