Gruber P. Convex and Discrete Geometry

140 Convex Bodies
7.5 Hadwiger’s Containment Problem
Consider the following
Problem 7.5. Specify (simple necessary and sufficient) conditions such that two
proper convex bodies satisfying these conditions have the property that one of the
bodies is contained in a congruent copy of the other body.
While, at present, a complete solution of this problem seems to be out of reach, there
are interesting contributions to it.
Hadwiger’s Sufficient Condition for Containment
As an application of the principal kinematic formula, we will prove the following
result of Hadwiger [461], where A and P stand for area and perimeter.
Theorem 7.11. Let C, D ∈ Cp(E 2 ).If
2π � A(C) + A(D) � − P(C)P(D) >0 and A(C) �= A(D),
then there is a rigid motion m such that either C ⊆ int mD or D ⊆ int mC.
Proof. Assume first that C and D are proper convex polygons. If C ∩ mD =∅, then
also bd C ∩ m bd D =∅. However, if C ∩ mD �= ∅, then there are two possibilities.
The first possibility is that bd C ∩ m bd D �= ∅. If we disregard a set of rigid motions
m of measure 0, then bd C ∩ m bd D consists of an even number of distinct points.
The second possibility is that bd C ∩m bd D =∅. Then C ⊆ int mD or D ⊆ int mC.
Suppose now that no congruent copy of C is contained in int D and, similarly,
with C and D exchanged. In terms of the Euler characteristic this implies the
following:
χ(C ∩ mD) = 0 ⇒ χ(bd C ∩ m bd D) = 0,
χ(C ∩ mD) = 1 ⇒ χ(bd C ∩ m bd D) ≥ 2
for all m ∈ M with a set of exceptions of measure 0. Hence
χ(bd C ∩ m bd D) ≥ 2 χ(C ∩ mD)
for all m ∈ M with a set of exceptions of measure 0. Thus,
�
�
χ(bd C ∩ m bd D) dµ(m) ≥ 2 χ(C ∩ mD) dµ(m),
M
or, by the principal kinematic formula 7.10 for d = 2 and the definition of Wi on
L(C),
M