Unit Disks With Respect To Other Metrics
One also considers unit disks with respect to other metrics. For instance, with the taxicab metric and the Chebyshev metric disks look like squares (even though the underlying topologies are the same as the Euclidean one).
The area of the Euclidean unit disk is π 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, 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.
Read more about this topic: Unit Disk
Famous quotes containing the words unit and/or respect:
“During the Suffragette revolt of 1913 I ... [urged] that what was needed was not the vote, but a constitutional amendment enacting that all representative bodies shall consist of women and men in equal numbers, whether elected or nominated or coopted or registered or picked up in the street like a coroner’s jury. In the case of elected bodies the only way of effecting this is by the Coupled Vote. The representative unit must not be a man or a woman but a man and a woman.”
—George Bernard Shaw (1856–1950)
“A common and natural result of an undue respect for law is, that you may see a file of soldiers, colonel, captain, corporal, privates, powder-monkeys, and all, marching in admirable order over hill and dale to the wars, against their wills, ay, against their common sense and consciences, which makes it very steep marching indeed, and produces a palpitation of the heart.”
—Henry David Thoreau (1817–1862)