Gruber P. Convex and Discrete Geometry

170 Convex Bodies
Note that �st
: �C →�� C is not continuous. Consider, for example, the sequence of line
1n
segments o, , 1 ∈ C(E2 ), n = 1, 2,... Clearly,
� � ��
1
o, , 1 →[o,(0, 1)] as n →∞,
n
but for Steiner symmetrization in the first coordinate axis we have that
� � �� � � �� ��
1
1
st o, , 1 = o, , 0 →{o} �= 0, −
n n 1
� �
, 0,
2
1
��
= st [o,(0, 1)] .
2
Proof. We may assume that o ∈ H. LetL be the line through o orthogonal to the
hyperplane H.
(i) The proof that st C is compact is left to the reader. To show that st C is convex,
let x, y ∈ st C. Consider the convex trapezoid
T = conv �� C ∩ (L + x) � ∪ � C ∩ (L + y) �� ⊆ C.
Since x, y ∈ st T and since st T is also a convex (see Fig. 9.2) trapezoid and is
contained in st C, it follows that [x, y] ⊆st T ⊆ st C.
(ii) Trivial.
(iii) Let x ∈ st C, y ∈ st D or, equivalently, x = h + l, y = k + m, where
h, k ∈ H and l, m ∈ L are such that �l� ≤ 1 2 length � C ∩ (L + x) � and �m� ≤ 1 2
length � D ∩ (L + y) � . Then x + y = (h + k) + (l + m) where h + k ∈ H (note that
o ∈ H) and l + m ∈ L (note that o ∈ L), where
�l + m� ≤�l�+�m�
≤ 1�
� � � ��
length C ∩ (L + x) + length D ∩ (L + y)
2
= 1
2 length � C ∩ (L + x) + D ∩ (L + y) �
≤ 1
2 length � (C + D) ∩ (L + x + y) � .
Hence x + y ∈ st(C + D).
T
H
y
x
st H T
Fig. 9.2. Steiner symmetrization preserves convexity