v2009.01.01 - Convex Optimization

44 CHAPTER 2. CONVEX GEOMETRY
2.1.9 inverse image
While epigraph and sublevel sets (3.1.7) of a convex function must be convex,
it generally holds that image and inverse image of a convex function are not.
Although there are many examples to the contrary, the most prominent are
the affine functions:
2.1.9.0.1 Theorem. Image, Inverse image. [266,3]
Let f be a mapping from R p×k to R m×n .
The image of a convex set C under any affine function (3.1.6)
is convex.
The inverse image 2.8 of a convex set F ,
f(C) = {f(X) | X ∈ C} ⊆ R m×n (24)
f −1 (F) = {X | f(X)∈ F} ⊆ R p×k (25)
a single- or many-valued mapping, under any affine function f is
convex.
⋄
In particular, any affine transformation of an affine set remains affine.
[266, p.8] Ellipsoids are invariant to any [sic] affine transformation.
Each converse of this two-part theorem is generally false; id est, given
f affine, a convex image f(C) does not imply that set C is convex, and
neither does a convex inverse image f −1 (F) imply set F is convex. A
counter-example is easy to visualize when the affine function is an orthogonal
projector [287] [215]:
2.1.9.0.2 Corollary. Projection on subspace. 2.9 (1809) [266,3]
Orthogonal projection of a convex set on a subspace or nonempty affine set
is another convex set.
⋄
Again, the converse is false. Shadows, for example, are umbral projections
that can be convex when the body providing the shade is not.
2.8 See Example 2.9.1.0.2 or Example 3.1.7.0.2 for an application.
2.9 For hyperplane representations see2.4.2. For projection of convex sets on hyperplanes
see [324,6.6]. A nonempty affine set is called an affine subset2.3.1. Orthogonal
projection of points on affine subsets is reviewed inE.4.