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
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