Gruber P. Convex and Discrete Geometry

3 Convex Sets, Convex Bodies and Convex Hulls 41
theorems of Helly, Radon and, again, Carathéodory, together with some applications.
As an example, where the notion of convexity is used in an analytic context to clarify
or to describe a situation, we present Hartogs’ theorem on power series in d complex
variables.
3.1 Basic Concepts and Simple Properties
We begin with the definitions of convex sets and convex bodies and investigate convex
hulls, including Carathéodory’s theorem. Then we consider convex cones and
prove a simple decomposition result.
Convex Sets and Convex Bodies
The earliest explicit mention of the notion of convexity seems to be in the first four
axioms in the book On the Sphere and Cylinder of Archimedes [35]. The third and
the fourth axiom are as follows:
3. Similarly also there are certain finite surfaces, not in a plane themselves but having
their extremities in a plane, and such that they will either lie wholly on the same side
of the plane containing their extremities or will have no part on the other side.
4. I call convex in the same direction surfaces such that, if any two points on them
are taken, either the straight lines between the points all fall upon the same side of
the surface, or some fall on one and the same side while others fall along the surface
itself, but none falls on the other side.
Archimedes thus actually gives two definitions of a convex surface. The first is by
means of support properties, the second is the common one by means of line segments
which are contained on the same side of the surface. In the sequel, we state
the second definition for sets instead of surfaces in the usual form and show the
equivalence of the two definitions later in Theorem 4.2.
AsetC in E d is convex if it has the following property:
It is strictly convex if it is closed and
(1 − λ)x + λy ∈ C for x, y ∈ C, 0 ≤ λ ≤ 1.
(1 − λ)x + λy ∈ int C for x, y ∈ C, x �= y, 0