Optimization and Control Theory Department: 2008-12-09 Seminar

prof. dr hab. Tadeusz Rzeżuchowski (The Faculty of Mathematics and Information Science at the Warsaw University of Technology)

Demyanov metric in the family of convex bounded sets

The seminar will be held at 9.00 in A1-33.

Let $f$ be a linear and continuous functional $f$ and $A$ a convex bounded subset of a linear space $X$ . We call the set $\{x\in A:f(x)=\sup_{y\in A}f(y)\}$ the face of $A$ supported by the function $f$ and denote it by $H_fA$ . If $\{f\in X^*:H_fA\ne\emptyset\}=\{f\in X^*:H_fB\ne\emptyset\}$ , we define the Demyanov distance between the sets $A$ and $B$ as $\sup_f d_H(H_fA,H_fB)$ , where $d_H$ is the usual Hausdorff metric.