15 Convex functions and sets[16V]
We will now discuss convexity. For simplicity, all results are presented using \({\mathbb {R}}^ n\) as domain; but most results hold more in general in a generic vector space.
15.1 Convex sets
Topology
See also exercises [122], [130] and [132].
Projection, separation
15.2 Convex function
Convex functions enjoy a lot of interesting properties, this one below is just a small list.
... equivalent definitions
Properties
The following is a list of properties for convex functions \(f:C→ℝ\) with \(C⊆ ℝ^ n\). Obviously these properties also apply when \(n=1\); but when \(n=1\) proofs are usually easier, see the next section.
15.3 Real case
Let \(I⊂ ℝ\), then \(I\) is convex if and only it is an interval (see [0S0]). In the following we will consider \(f:I→ℝ\) where \(I=(a,b)\) is an open interval.
Convexity and derivatives
See also the exercise [1BF] for the relationship between integral and convexity.
Convex functions with extended values
We consider convex functions that can also take on value \(+∞.\) Let \(I\) be an interval.
15.4 Additional properties and exercises
See also exercise [1C3].
Distance function
Strictly convex functions and sets
- E7
One wonders now, what if \(f\) is strictly convex?
- E10
[ [19J]]