A Borsuk-Ulam theorem for digital images

The Borsuk-Ulam theorem states that a continuous function f : S → R has a point x ∈ S with f(x) = f(−x). We give an analogue of this theorem for digital images, which are modeled as discrete spaces of adjacent pixels equipped with Z-valued functions. In particular, for a concrete two-dimensional rectangular digital image whose pixels all have an assigned “brightness” function, we prove that there must exist a pair of opposite boundary points whose brightnesses are approximately equal. This theorem applies generally to any integervalued function on an abstract simple graph. We also discuss generalizations to digital images of dimension 3 and higher. We give some partial results for higher dimensional images, and show a counter example which demonstrates that the full results obtained in lower dimensions cannot hold generally.