Convex Function
A function is convex when
for all and : the graph lies below every chord.
Definition 3.34 (Convex Function)
A function is called convex if , , we have
Supporting lines
A convex function is the supremum of all the lines below it: for every there exist with
To see this, take and with , so that . Convexity gives
Hence the slopes satisfy for all , so for every , and completes the supporting line.
Related
Stated in
- Definition 3.34 (Convex Function)§3.3 Inequalities