Elementary Properties of Extremal Length
The extremal length satisfies a few simple monotonicity properties. First, it is clear that if, then . Moreover, the same conclusion holds if every curve contains a curve as a subcurve (that is, is the restriction of to a subinterval of its domain). Another sometimes useful inequality is
This is clear if or if, in which case the right hand side is interpreted as . So suppose that this is not the case and with no loss of generality assume that the curves in are all rectifiable. Let satisfy for . Set . Then and, which proves the inequality.
Read more about this topic: Extremal Length
Famous quotes containing the words elementary, properties and/or length:
“Listen. We converse as we live—by repeating, by combining and recombining a few elements over and over again just as nature does when of elementary particles it builds a world.”
—William Gass (b. 1924)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“A playwright ... is ... the litmus paper of the arts. He’s got to be, because if he isn’t working on the same wave length as the audience, no one would know what in hell he was talking about. He is a kind of psychic journalist, even when he’s great.”
—Arthur Miller (b. 1915)