Gruber P. Convex and Discrete Geometry

(5)
�
C
v(x) dx = lim
�
n→∞
C
vn(x) dx = lim
n→∞ αvn(o) = αv(o)
12 Special Convex Bodies 225
by Proposition (3). For the notions of stochastic convergence and uniform integrability
and for the first equality in (5) see, e.g. Bauer [82], Sects. 20, 21.
Finally, let a harmonic function w be defined by
w(x) = 1 +�p�v(x) for x ∈ E 3 \{p}
and let B be the ball �p�B 3 . Since o ∈ int C by (2), our choice of p shows that
B ⊆ C. Thus
(6) αw(o) = α � 1 +�p�v(o) � �
= 0 =
�
� �
1 +�p�v(x) dx = w(x) dx
by the definitions of w and v, (3) applied to u = 1 and Proposition (5). The mean
value property of harmonic functions implies that
�
(7) w(x) dx = γw(o) = 0,
B
where γ > 0 is a suitable multiplicative constant, see Wermer [1018], Appendix.
From (6) and (7) we conclude that
�
�
�
�
0 = w(x) dx = w(x) dx + w(x) dx = w(x) dx.
C
B
Since w(x) > 0forx ∈ C\B, it follows that C\B has measure 0. Noting that
C = cl int C by assumption, this means that C\B =∅,orC ⊆ B. The reverse
inclusion being obvious, C = B follows. ⊓⊔
C
C\B
C\B
12.3 Blaschke’s Characterization of Ellipsoids and Its Applications
Ellipsoids and Euclidean spaces play an important role in many branches of mathematics.
Among these are convex and differential geometry, the local theory of
normed spaces, functional analysis, approximation, operator and potential theory,
dynamical systems, combinatorial optimization and mechanics. The first characterizations
of ellipsoids in convex geometry go back to Brunn and Blaschke, and at
present there is still interest for these characterizations. One may distinguish between
characterizations based on:
Affine and projective transformations
Sections
Projections and illuminations
Extremal properties
Other geometric and analytic properties
C