Gruber P. Convex and Discrete Geometry

image of
the sun
screen
6 Mixed Volumes and Quermassintegrals 81
Fig. 6.1. Camera obscura
�
sun
diaphragm
A similar question due to Tycho Brahe is about the form of the image of the sun on
the screen of a camera obscura, depending on the shape of the diaphragm (Fig. 6.1).
Implicitly using Minkowski addition, Kepler [575] solved Brahe’s problem by showing
– in our terminology – the following: the image of the sun is of the form
D + λB 2 , where D is a convex disc, a translate of the diaphragm. See Scriba and
Schreiber [923]. A similar phenomenon can be observed in sunshine at noon under
a broad-leaved tree: the image of the sun on the ground consists of numerous rather
round figures of the form P + λE, where P is approximately of polygonal shape,
not necessarily convex, and E an ellipse. See Schlichting and Ucke [890].
Properties of Minkowski Addition
The following simple properties of Minkowski addition will be used frequently.
Proposition 6.1. Let C, D ∈ C and λ ∈ R. Then C + D, λC ∈ C.
Proof. We consider only C + D. To show the convexity of C + D,letu + x,v+ y ∈
C + D where u,v ∈ C, x, y ∈ D, and let 0 ≤ λ ≤ 1. Then
(1 − λ)(u + x) + λ(v + y) = � (1 − λ)u + λv � + � (1 − λ)x + λy � ∈ C + D
by the convexity of C and D. Hence C+D is convex. It remains to show that C+D is
compact. The Cartesian product C × D ={(x, y) : x ∈ C, y ∈ D} ⊆E d ×E d = E 2d
is compact in E 2d . Being the image of the compact set C × D under the continuous
mapping (x, y) → x + y of E 2d onto E d ,thesetC + D is also compact. ⊓⊔
The following result relates addition and multiplication with positive numbers of
convex bodies to addition and multiplication with positive numbers of the corresponding
support functions.
Proposition 6.2. Let C, D ∈ C and λ ≥ 0. Then
Proof. Again, we consider only C + D:
hC+D = hC + h D, hλC = λhC.
hC+D(u) = sup{u · (x + y) : x ∈ C, y ∈ D}
= sup{u · x : x ∈ C}+sup{u · y : y ∈ D}
= hC(u) + h D(u) for u ∈ E d . ⊓⊔