On a discrete version of the antipodal theorem
Volume 151 / 1996
Fundamenta Mathematicae 151 (1996), 189-194 DOI: 10.4064/fm-151-2-189-194
The classical theorem of Borsuk and Ulam  says that for any continuous mapping $f: S^k → ℝ^k$ there exists a point $x ∈ S^k$ such that f(-x) = f(x). In this note a discrete version of the antipodal theorem is proved in which $S^k$ 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 $inf_x ||f(x)-f(-x)||$ which were previously known (as far as the author knows) only for f linear (cf. ).