Gruber P. Convex and Discrete Geometry

76 Convex Bodies
Hence x ∈ conv ext D. Ifx ∈ int D, choose y, z ∈ bd D such that x ∈[y, z]. Bythe
case just considered, y, z ∈ conv ext D and thus x ∈ conv ext D. This concludes the
proof of (1).
The second step is to show the following:
(2) Let x ∈ ext C. Then C\{x} is convex.
If (2) did not hold, there are points y, z ∈ C\{x} such that [y, z] �⊆ C\{x}. Since
[y, z] ⊆C by the convexity of C, this can hold only if x ∈[y, z]. Since x �= y, z,
the point x is not extreme, a contradiction. The proof of (2) is complete.
Finally, (1) and (2) yield the theorem. ⊓⊔
Remark. The following refinement of Theorem 5.5 is due to Straszewicz [973]:
C = cl conv exp C, where exp C is the set of all exposed points of C. These are the
points x ∈ C such that {x} =C ∩ H for a suitable support hyperplane of C at x.
Clearly, each exposed point is extreme, but the converse does not hold generally. It
is not difficult to see that the apex of a pointed closed convex cone C is an exposed
and thus an extreme point of C.
Maxima of Convex Functions
A simple yet useful application of the above result is as follows:
Theorem 5.6. Let C ∈ C and let f : C → R be a continuous convex function. Then
f attains its maximum m at an extreme point of C.
Proof. Let x ∈ C be such that f (x) = m. Theorem 5.5 and Lemma 3.1 show that
x = λ1x1 +···+λnxn with suitable x1,...,xn ∈ ext C and λ1,...,λn > 0, where
λ1 +···+λn = 1. Jensen’s inequality then yields the following:
m = f (x) ≤ λ1 f (x1) +···+λn f (xn) ≤ (λ1 +···+λn)m = m.
Thus equality holds throughout. Noting that λ1,...,λn > 0, it then follows that
f (x1) =···= f (xn) = m. ⊓⊔
5.3 Birkhoff’s Theorem on Doubly Stochastic Matrices
Doubly stochastic d × d matrices have attracted a lot of interest, for example as the
matrices of transition probabilities of discrete Markov chains or in the context of van
der Waerden’s conjecture on permanents of doubly stochastic matrices.
Using the notions of convex hull and extreme point, we give a precise description
of the set of doubly stochastic d × d matrices due to Birkhoff.
For more information on van der Waerden’s conjecture, see Egorychev [291].
For general information on doubly stochastic matrices compare Minc [729,730] and
Seneta [926].