CECM Colloquium

**Wednesday September 19, 2001
in K9509, SFU
**

3:30 - 4:20

Dr Kazimierz Goebel

Lublin

**Title:**
"Minimal displacement and optimal retraction problems in
Banach spaces"

**Abstract:**
Let (X,|| ||) be an infinite dimensional Banach space with unit ball B and unit sphere S. It is known that Brouwer's Fixed Point Theorem "strongly fails" in this setting. This means that:

- A. There are lipschitzian mappings T:B->B without fixed points and, even more, such that inf || x-Tx|| =: d(T) > 0.
- B. The unit sphere S is a lipschitzian retract of B meaning that there is a lipschitzian mapping (a retraction) R:B->S such that T |_S =Id.$
- C. The unit sphere S is contractible to a point via lipschitzian homotopy.