J. Leray defined a local fixed
point index for functions defined in what he called convexoid spaces. From the
standpoint of analysis, the most important example of a convexoid space is a
compact subset C ⊂ X,X a locally convex topological vector space, such that
C = ∪i=1nCi, where Ci are compact, convex subsets of X or a homeomorphic image
of such a C. In this paper a simple geometric approach is given (see Lemma 2 below)
by means of which a fixed point index can be defined for functions with domain in a
class of spaces ℱ which contains the spaces C mentioned above and also the compact
metric ANR’s. The usual properties of the fixed point index are established, and it is
shown that they axiomatically determine the index for the class of spaces
ℱ.
