Gruber P. Convex and Discrete Geometry

54 Convex Bodies
in one of the two closed halfspaces determined by H. In this case, we denote the
halfspace containing C by H − , the other one by H + . H − is called a support halfspace
of C at x. In general, we represent H in the form H ={z : u · z = u · x}, where
u is a normal (unit) vector of H pointing into H + . Then H − ={z : u · z ≤ u · x}
and H + ={z : u · z ≥ u · x}. u is an exterior normal (unit) vector of H, ofC or of
bd C at x. Note that H may not be unique. The intersection C ∩ H is the support set
of C determined by H or, with exterior normal (unit) vector u.
In Theorems 1.2 and 2.3, it was shown that a convex function has affine support
at each point in the interior of its domain of definition. The following theorem is the
corresponding result for convex sets.
Theorem 4.1. Let C ⊆ E d be a closed convex set. For each x ∈ bd C, there is a
support hyperplane HC(x) of C at x, not necessarily unique. If C is compact, then
for each vector u ∈ E d \{o}, there is a unique support hyperplane HC(u) of C with
exterior normal vector u.
Let S d−1 denote the Euclidean unit sphere in E d .
Proof. First, the following will be shown.
(1) Let y ∈ E d \C. Then the hyperplane H through pC(y) ∈ bd C, orthogonal
to y − pC(y), supports C at pC(y).
It is sufficient to show that H separates y and C. If this does not hold, there is a point
z ∈ C which is not separated from y by H. Then the line segment [pC(y), z] ⊆C
contains a point of C which is closer to y than pC(y). This contradicts the definition
of pC(y) and thus concludes the proof of (1).
Next we claim the following.
(2) Let Hn ={z : un · z = xn · un} be support hyperplanes of C at the points
xn ∈ bd C, n = 1, 2,... Assume that un → u ∈ E d \{o} and xn → x (∈
bd C) as n →∞. Then H ={z : u · z = u · x} is a support hyperplane of
C at x.
Clearly, x ∈ H. It is sufficient to show that C ⊆ H − .Letz ∈ C. Then un ·z ≤ un · xn
for n = 1, 2,... Letting n →∞, we see that u · z ≤ u · x, or z ∈ H − , concluding
the proof of (2).
For the proof of the first assertion in the theorem, choose points yn ∈ E d \C, n =
1, 2,...,such that yn → x. By Lemma 4.1, xn = pC(yn)(∈ bd C) → x = pC(x).
Proposition (1) shows that, for n = 1, 2,..., there is a support hyperplane of C at
xn,sayHn ={z : un · z = un · xn}, where un ∈ S d−1 . By considering a subsequence
and re-numbering, if necessary, we may suppose that un → u ∈ S d−1 ,say.An
application of (2) then implies that H ={z : u · z = u · x} is a support hyperplane of
C at x.
To see the second assertion, note that the compactness of C implies that u · x =
sup{u · z : z ∈ C} for a suitable x ∈ C. Clearly, x ∈ bd C and H ={z : u · z = u · x}
is a support hyperplane of C (at x) with exterior normal vector u. ⊓⊔