v2007.09.13 - Convex Optimization

44 CHAPTER 2. CONVEX GEOMETRY2.1.9 inverse imageWhile 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 arethe affine functions:2.1.9.0.1 Theorem. Image, Inverse image. [228,3] [46,2.3.2]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.[228, p.8] Ellipsoids are invariant to any [sic] affine transformation.Each converse of this two-part theorem is generally false; id est, givenf affine, a convex image f(C) does not imply that set C is convex, andneither does a convex inverse image f −1 (F) imply set F is convex. Acounter-example is easy to visualize when the affine function is an orthogonalprojector [247] [181]:2.1.9.0.2 Corollary. Projection on subspace. [228,3] 2.9Orthogonal projection of a convex set on a subspace is another convex set.⋄Again, the converse is false. Shadows, for example, are umbral projectionsthat can be convex when the body providing the shade is not.2.8 See2.9.1.0.2 for an example.2.9 The corollary holds more generally for projection on hyperplanes (2.4.2); [277,6.6]hence, for projection on affine subsets (2.3.1, nonempty intersections of hyperplanes).Orthogonal projection on affine subsets is reviewed inE.4.0.0.1.