Really just the triangle inequality for p-norms:
For any p>=1, a_{1},...,a_{n}≥0 and b_{1},...,b_{n}≥0,

((a_{1}+b_{1})^{p} + ... + (a_{n}+b_{n})^{p})^{1/p} ≤
(a_{1}^{p}+...+a_{n}^{p})^{1/p} +
(b_{1}^{p}+...+b_{n}^{p})^{1/p}

A version with integrals (instead of sums) exists too, of course.

A proof of Minkowski's inequality follows from some trickery with Hölder's inequality.