Editing Unit disk (section)

== Unit disks with respect to other metrics ==
[[File:Unit disc metrics.svg|thumb|From top to bottom: open unit disk in the [[Euclidean metric]], [[taxicab metric]], and [[Chebyshev metric]].]]
One also considers unit disks with respect to other [[Metric (mathematics)|metric]]s. For instance, with the [[taxicab geometry|taxicab metric]] and the [[Chebyshev distance|Chebyshev metric]] disks look like squares (even though the underlying [[topological space|topologies]] are the same as the Euclidean one).

The area of the Euclidean unit disk is [[pi|&pi;]] and its [[perimeter]] is 2π. In contrast, the perimeter (relative to the taxicab metric) of the unit disk in the taxicab geometry is 8. In 1932, [[Stanisław Gołąb]] proved that in metrics arising from a [[norm (mathematics)|norm]], the perimeter of the unit disk can take any value in between 6 and 8, and that these extremal values are obtained if and only if the unit disk is a regular [[hexagon]] or a [[parallelogram]], respectively.