Gruber P. Convex and Discrete Geometry

224 Convex Bodies
The integral here may be considered as the convolution of the locally integrable
function 1/�x� on E 3 \{o} with the characteristic function of C which is bounded,
measurable and has compact support. It thus defines a bounded continuous function
on all of E 3 . Then
(2) o ∈ int C,
since otherwise the function α/�y� is unbounded on E 3 \C and thus the above
integral is unbounded by (1), a contradiction.
A real function u on a domain in E 3 of class C 2 which satisfies the Laplace
equation
�u = ∂2 u
∂x 2 1
is harmonic. We now show the following:
+ ∂2 u
∂x 2 2
+ ∂2 u
∂x 2 3
= 0
(3) Let u be a harmonic function on a neighbourhood of C. Then
�
C
u(x) dx = α u(o),
where α is as in (1). Modify u outside C to be of class C2 with compact support S,
preserving the harmonicity on and near C. Since u is of class C2 and has compact
support, a potential theoretic result shows that
(4) u(x) =− 1
4π
�
S
�u(y)
�x − y� dy for x ∈ E3 ,
see Wermer [1018], p.13. Thus,
�
u(x) dx =−
C
1
� �
4π
�
�u(y)
1
dydx =−
�x − y� 4π
C S
=−
S
1
�
�u(y)
4π
α
dy = α u(o)
�y�
S
� �
�u(y)
C
dx
�
dy
�x − y�
by (4), Fubini’s theorem, (1) and (4) again, concluding the proof of (3).
Now, noting (2), choose p ∈ bd C closest to o and take points p1, p2, ···∈E 3 \C
with pn → p. Each of the functions vn defined by
vn(x) = �x�2 −�pn� 2
�x − pn� 3
for x ∈ E 3 \{pn}
is harmonic on C and vn(x) → v(x) for x ∈ C \{p}, where v is a harmonic function
defined by
v(x) = �x�2 −�p�2 �x − p�3 for x ∈ E 3 \{p}.
The sequence � vn(·) � converges pointwise and thus stochastically to v(·) on C\{p}
and is uniformly integrable on C. Thus