Chapter 3 Geometry of convex functions - Meboo Publishing ...

232 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
{a T z 1 + b 1 | a∈ R}
supa T pz i + b i
i
{a T z 2 + b 2 | a∈ R}
{a T z 3 + b 3 | a∈ R}
a
{a T z 4 + b 4 | a∈ R}
{a T z 5 + b 5 | a∈ R}
Figure 72: Pointwise supremum of convex functions remains convex; by
epigraph intersection. Supremum of affine functions in variable a evaluated
at argument a p is illustrated. Topmost affine function per a is supremum.
a positively homogeneous function of direction a whose range contains ±∞.
[244, p.135] For each z ∈ Y , a T z is a linear function of vector a . Because
σ Y (a) is a pointwise supremum of linear functions, it is convex in a
(Figure 72). Application of the support function is illustrated in Figure 29a
for one particular normal a . Given nonempty closed bounded convex sets Y
and Z in R n and nonnegative scalars β and γ [364, p.234]
σ βY+γZ (a) = βσ Y (a) + γσ Z (a) (532)
3.5.0.0.4 Exercise. Level sets.
Given a function f and constant κ , its level sets are defined
L κ κf {z | f(z)=κ} (533)
Give two distinct examples of convex function, that are not affine, having
convex level sets.