Gruber P. Convex and Discrete Geometry

4 Support and Separation 55
Remark. The proofs of Theorem 2.3 on affine support of convex functions and of
the corresponding Theorem 4.1 on support hyperplanes of closed convex sets are
essentially different. The former proof is based on the linearity structure of E d and
can be extended easily to infinite dimensions by transfinite induction. The latter proof
rests on the Euclidean structure of E d and on compactness and its extension to infinite
dimensions is not of much interest.
Results on convex functions correspond, in many cases, to results on convex sets
or convex bodies and vice versa. Quite often it is possible to transform a result on
convex functions into a result on convex sets or convex bodies. Examples deal with
support, separation, and differentiability properties.
Characterization of Convex Sets by Support Properties
The definition of a support hyperplane of C at a point x ∈ bd C still makes sense
if the assumption that C is convex is omitted. Thus we may speak of a support
hyperplane ofaclosedsetinE d at a boundary point. The characterizations of convex
functions by support properties in Theorems 1.3 and 2.4 correspond to the following
result.
Theorem 4.2. Let C ⊆ E d be closed and let int C �= ∅. Then the following are
equivalent:
(i) C is convex.
(ii) C has a support hyperplane HC(x) at each point x ∈ bd C.
Proof. (i)⇒(ii) This follows from Theorem 4.1.
(ii)⇒(i) The intersection of any family of convex sets is also convex. Thus it
suffices to prove that
(3) C = �
H − C (x).
x∈bd C
Since C is contained in the set on the right-hand side, it remains to show the
following: Let z ∈ Ed \C. Then z �∈ H − C (x) for a suitable x ∈ bd C. To see this,
let y ∈ int C and choose x ∈ [y, z] ∩bd C. By (ii) there is a support halfspace
H − C (x) of C at x. Then y ∈ int C ⊆ C ⊆ H − C (x) and therefore z �∈ H − C (x). ⊓⊔
Corollary 4.1. Let C ⊆ Ed be closed and convex. Then
C = �
How to Specify a Convex Body
x∈bd C
H − C (x).
While the definitions of convex sets and convex bodies are extremely simple, it is a
highly non-trivial task to specify an arbitrary convex body so that one may obtain,
from this specification, important information about the body, for example analytic