Minkowski's Inequality

|a + b|_p ≤ |a|_p + |b|_p — the triangle inequality in every L^p.

The idea

Theorem (Minkowski's inequality).

For $p \geq 1$ and any sequences $a$ and $b$, $\|a + b\|_p \;\leq\; \|a\|_p + \|b\|_p, \qquad \text{where } \|a\|_p = \left(\sum_i |a_i|^{p}\right)^{1/p}.$ For $p > 1$, equality holds only when one of the two sequences is a nonnegative multiple of the other.

This is the triangle inequality for the $p$-norm: the size of a sum is at most the sum of the sizes. The figure shows the triangle for two vectors $x$ and $y$: the two legs have lengths $\|x\|$ and $\|y\|$, and the direct side, of length $\|x + y\|$, is never longer than the two legs together. Equality occurs when the two legs point the same way, so the broken path is itself straight.

The hypothesis $p \geq 1$ is essential. For $0 < p < 1$ the inequality fails (for nonnegative sequences it reverses), and $\|\cdot\|_p$ is not a norm, because the definition of a norm requires exactly this triangle inequality. Minkowski's inequality is the fact that qualifies $\|\cdot\|_p$ as a norm.

Ways to work on it

• Walkthrough. Verify the inequality on a concrete pair of vectors, then identify the exponents for which it holds.
• Proof. Derive the L^p triangle inequality from Hölder's inequality.
• Practice. Check the inequality numerically on random integer vectors.
• Hardest. Compute the gap between the two sides of the inequality for random vectors.

Not sure where to start? Take the ten-question placement test.