Lipschitz Differentiability of the Supporting Function from a Geometric Viewpoint, the Nearest and the Farthest Points

We consider a supporting condition for a convex compact set in Rn. This condition is typical and it is closely related to the local Lipschitz differentiability of the supporting function. With the help of the considered condition we prove a local Lipschitz condition for supporting elements and for nearest and farthest points of a convex compact set. Some applications for numerical algorithms are discussed.