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 $\mathbb{R}^n$. 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.