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:
“When the Devil quotes Scriptures, it’s not, really, to deceive, but simply that the masses are so ignorant of theology that somebody has to teach them the elementary texts before he can seduce them.”
—Paul Goodman (1911–1972)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“At length to hospital
This man was limited,
Where screens leant on the wall
And idle headphones hung.
Since he would soon be dead
They let his wife come along
And pour out tea, each day.”
—Philip Larkin (1922–1986)