The classical theorem of Borsuk and Ulam [2] says that for any continuous mapping there exists a point such that f(-x) = f(x). In this note a discrete version of the antipodal theorem is proved in which is replaced by the set of vertices of a high-dimensional cube equipped with Hamming’s metric. In place of equality we obtain some optimal estimates of which were previously known (as far as the author knows) only for f linear (cf. [1]).
Let be a closed convex subset of a complete convex metric space . In this paper a class of selfmappings on , which satisfy the nonexpansive type condition below, is introduced and investigated. The main result is that such mappings have a unique fixed point.
Let (G,τ) be a Hausdorff Abelian topological group. It is called an s-group (resp. a bs-group) if there is a set S of sequences in G such that τ is the finest Hausdorff (resp. precompact) group topology on G in which every sequence of S converges to zero. Characterizations of Abelian s- and bs-groups are given. If (G,τ) is a maximally almost periodic (MAP) Abelian s-group, then its Pontryagin dual group is a dense -closed subgroup of the compact group , where is the group G with the discrete...