The fixed-point property of convex compacta in nonlocally convex spaces
by
Tadeusz Dobrowolski
Pittsburg State University

In 1930, J. Schauder published a paper in which he claimed that every convex compact subset X of a metric linear space had the fixed-point property. His proof, however, contained a gap. The gap occured while Schauder was verifying the so-called simplicial approximation property, a property that was later formally introduced and investigated in a book by Kalton, Peck, and Roberts. Consequently, the question of whether every convex compact subset of a metric linear space has the fixed-point property was put in the Scottish Book as Problem 54; it became known as Schauder's Conjecture. Since then many partial results were obtained, but the general case went unsettled. Until recently, the most transparent fixed-point theorem for convex sets, in textbooks referred to as Schauder-Tychonoff Theorem, has been phrased as follows: Every convex compact subset of a locally convex topological vector space has the fixed-point property.

The adjective ``locally convex'' can be removed from the above statement. This surprising result, obtained by R. Cauty in 1999, finally solved the Schauder Conjecture. Analyzing the argument of Cauty, we have isolated two of its nonoverlapping ingredients; one of them is of purely topological nature, while the other involves the convexity of X. The topological ingredient provides a ``nice'' parametrization of the compactum X (which can be also viewed as a ``nice'' resolution of X). Further investigation of that parametrization has led us to the verification of the simplicial approximation property of X. This confirms that the original approach of Schauder can be corrected.

For the locally convex case, the fixed-point property of X easily yields the fixed-point property for convex-valued maps of X (by the use of partitions of unity). This cannot be copied in the nonlocally convex case. However, a certain technical result, which relies on Cauty's proof, can be utilized to extend the fixed-point property for such multivalued maps.

Date received: February 22, 2002