v2009.01.01 - Convex Optimization

210 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
f(z)
A
z 2
z 1
C
H−
H+
a
{z ∈ R 2 | a T z = κ1 }
{z ∈ R 2 | a T z = κ2 }
{z ∈ R 2 | a T z = κ3 }
Figure 63: Cartesian axes in R 3 and three hyperplanes intersecting convex set
C ⊂ R 2 reproduced from Figure 24. Plotted with third dimension is affine set
A = f(R 2 ) a plane. Sequence of hyperplanes, w.r.t domain R 2 of an affine
function f(z)= a T z + b : R 2 → R , is increasing in direction of gradient a
(3.1.8.0.3) because affine function increases in normal direction (Figure 22).