Gruber P. Convex and Discrete Geometry

156 Convex Bodies
Let ϱ>0 be the maximum radius of a circular disc inscribed to B. Define the
inner parallel set of B at distance δ, where 0 ≤ δ ≤ ϱ, by:
(2) B−δ ={x ∈ B : x + δB 2 ⊆ B} ={x ∈ B : δ(x) ≥ δ}.
Clearly, B−δ is compact, and since B−δ + δB 2 ⊆ B for 0 ≤ δ ≤ ϱ, the Brunn–
Minkowski theorem for compact sets shows that
(3) A(B−δ) 1 2 + A(δB 2 ) 1 2 ≤ A � B−δ + δB 2� 1 1
2 ≤ A(B) 2 , and thus,
A(B−δ) ≤ � A(B) 1 2 − δπ 1 �2 2 for 0 ≤ δ ≤ ϱ.
Combining (2), Fubini’s theorem applied to the integral in (1), and (3), we obtain
the following:
�
V ≤ tan α
B
ϱ
0
�
δ(x) dx = tan α
0
ϱ
A(B−δ) dδ
�
� 1
≤ tan α A(B) 2 − δπ 1 �2dδ tan α � 1
2 =− A(B) 2 − δπ 1 � �
3�� ϱ
2
0
3π 1 2
tan α
=−
3π 1 � 1
A(B) 2 − ϱπ
2
1 �3 tan α A(B)
2 + 3 2
3π 1 2
≤ tan α A(B) 3 2
3π 1 2
where in the last inequality there is equality if and only if A(B) = ϱ 2 π, i.e. when B
coincides with its maximum inscribed circular disc, up to a set of measure 0. Thus
we have proved the following: The volume of a sand pile on B is bounded above by
tan α A(B) 3 2 /3π 1 2 and this bound can be attained, if at all, only when B is a circular
disc up to a set of measure 0. A simple check shows that, in fact, there is equality if
B is a circular disc – consider the circular cone with angle α with the horizontal. ⊓⊔
Equilibrium Capillary Surfaces
Let B be a compact body in E 2 bounded by a closed Jordan curve K of class C 2 .
Let E 2 be embedded into E 3 as usual (first two coordinates). Consider a vertical
cylindrical container with cross-section B and filled with water up to the level of B.
Due to forces in the surface of the water, the surface deviates from B close to the
boundary of the container. If u(x), x ∈ B ∪ K , describes this deviation, then the
following statements are well known (see Fig. 8.2):
(4) u is of class C 1 on B,ofclassC 2 on int B, and it is the unique such function
satisfying
grad u
div �
1 + (grad u) 2 � 1 = κ u on int B,
2
(grad u) · n
� 1 + (grad u) 2 � 1 2
,
= cos α on K.