Exercise
46
[19B] Topics:Distance function, convex sets. Prerequisites:[0R9],[17D].
Given \(A⊂ ℝ^ n\) a closed convex set, we define the distance function \(d_ A(x)\) as in [0R9]; let \(z∉ A\) and \(x^*\) the projection of \(z\) on \(A\) (i.e. the point of minimum distance in the definition of \(d_ A(z)\)). Having fixed \(v=(z-x^*)/|z-x^*|\), show that \(v∈∂ f(z)\); where \(∂ f\) is the subdifferential defined in [188].