We prove the following generalisation of Schauder's fixed point conjecture:
Let C_1,...,C_n be convex subsets of a Hausdorff topological vector space.
Suppose that the C_i are closed in C=C_1∪...∪ C_n. If f:C→ C is a
continuous function whose image is contained in a compact subset of C, then
its Lefschetz number Λ(f) is defined. If Λ(f)0, then f has
a fixed point.
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