Reverse Triangle Inequality

Turning Distance into a Lower Bound

The Triangle Inequality, recalled in the previous tutorial, gives an upper bound: the magnitude of a sum cannot exceed the sum of the magnitudes. Sometimes an upper bound is not what is needed. If two quantities nearly cancel, we may instead want to know how small their sum can be, or how much the magnitude of a number can change when the number is perturbed.

The reverse triangle inequality supplies this kind of information. It was established in Properties of Absolute Value: for real numbers \(u\) and \(v\), the difference between their magnitudes is no greater than the magnitude of their difference. We will use that result, rather than prove it again, to obtain useful lower bounds and magnitude estimates.

Because an absolute value is nonnegative, the reverse triangle inequality immediately implies both \(|u|-|v|\leq|u-v|\) and \(|v|-|u|\leq|u-v|\). Either inequality can be useful as a lower bound after rearranging. The order of the terms matters: the difference on the right is \(u-v\), but its magnitude is unchanged if the order is reversed.

A Lower Bound for the Magnitude of a Sum

To estimate a sum from below, compare one term with the negative of the other. The distance between \(u\) and \(-v\) is \(|u+v|\), so the reverse triangle inequality compares that sum with the difference of the terms’ magnitudes.

Proof. Apply the previously established reverse triangle inequality to the pair \(u\) and \(-v\). It gives

The basic properties of absolute value give \(|-v|=|v|\), and subtraction gives \(u-(-v)=u+v\). Substituting these identities into the inequality proves

Since \(|u|-|v|\leq\bigl||u|-|v|\bigr|\) and \(|v|-|u|\leq\bigl||u|-|v|\bigr|\), both stated one-sided lower bounds follow. \(\square\)

The absolute difference in the theorem is important. If one term has much larger magnitude than the other, the sum cannot be close to zero: a smaller term cannot cancel more than its own magnitude. If the two magnitudes are equal, however, this estimate only says \(|u+v|\geq0\), which is always true. In that case the terms may cancel completely, so a stronger lower bound cannot generally be expected from their magnitudes alone.

How Much Can a Magnitude Change?

The reverse triangle inequality also controls changes in absolute value. If a number \(x\) is close to a reference number \(a\), then \(|x|\) must be close to \(|a|\). More precisely, the change in magnitude is at most the distance between \(x\) and \(a\). The next result packages this observation as upper and lower bounds that can be used directly.

Proof. By the reverse triangle inequality applied to \(x\) and \(a\),

From \(|x|-|a|\leq r\), adding \(|a|\) to both sides gives \(|x|\leq|a|+r\). From \(|a|-|x|\leq r\), rearranging gives \(|a|-r\leq|x|\). Also \(|x|\geq0\), by the basic properties of absolute value. Thus \(|x|\) is at least both \(0\) and \(|a|-r\), which means

Combining the lower and upper bounds proves the result. \(\square\)

The maximum with \(0\) handles an important edge case. If \(r>|a|\), then \(|a|-r\) is negative, and that negative number is not a useful lower bound for an absolute value. The fact that \(|x|\geq0\) gives the meaningful lower bound instead. If \(r\leq|a|\), the estimate says that \(x\) cannot move so far from \(a\) that its magnitude drops below \(|a|-r\).

Using Distance Estimates to Control Expressions

The perturbation theorem is useful when a quantity is known to be close to a simpler reference value. One can estimate the magnitude of the complicated quantity by first estimating its distance from the reference, then adjusting the reference magnitude by at most that distance. This avoids expanding an expression or determining its sign.

For example, suppose \(t\) is close to \(-3\), and \(|t-(-3)|\leq0.2\). The theorem with \(a=-3\) and \(r=0.2\) gives \(2.8\leq|t|\leq3.2\). The negative sign of the reference value does not change its magnitude: \(|-3|=3\). A distance estimate works around positive and negative reference values in exactly the same way.

The lower-bound theorem for sums and the perturbation theorem are closely related uses of the same idea. In the first, we compare \(u\) with \(-v\), so their distance is the magnitude of a sum. In the second, we compare \(x\) with a reference value \(a\), so their distance measures how far the input has changed. In each case, the reverse triangle inequality turns a distance into information about magnitudes.

A common pitfall is to assume that a lower bound is an equality, or to use \(|u|-|v|\) as if it were always positive. The safe sum estimate is \(\bigl||u|-|v|\bigr|\leq|u+v|\). If the magnitudes are equal, the estimate gives only zero; if they are unequal, it gives a positive lower bound. Check which situation applies before drawing a conclusion about cancellation.