The reverse triangle inequality is an elementary consequence of the triangle inequality that gives lower bounds instead of upper bounds. For plane geometry the statement is:
- Any side of a triangle is greater than the difference between the other two sides.
In the case of a normed vector space, the statement is:
or for metric spaces, | d(y, x) − d(x, z) | ≤ d(y, z). This implies that the norm ||–|| as well as the distance function d(x, –) are Lipschitz continuous with Lipschitz constant 1, and therefore are in particular uniformly continuous.
The proof for the reverse triangle uses the regular triangle inequality, and :
Combining these two statements gives:
Read more about this topic: Triangle Inequality
Famous quotes containing the words reverse and/or inequality:
“They shall beat their swords into plowshares, and their spears into pruninghooks: nation shall not lift up sword against nation, neither shall they learn war any more.”
—Bible: Hebrew Isaiah, 2:4.
The words reappear in Micah 4:3, and the reverse injunction is made in Joel 3:10 (”Beat your plowshares into swords ...”)
“All the aspects of this desert are beautiful, whether you behold it in fair weather or foul, or when the sun is just breaking out after a storm, and shining on its moist surface in the distance, it is so white, and pure, and level, and each slight inequality and track is so distinctly revealed; and when your eyes slide off this, they fall on the ocean.”
—Henry David Thoreau (1817–1862)