A proof of the Minkowski inequalities based on convex homogeneous functions

The triangle inequality for p-norms, also known as the Minkowski inequality, is often proven with algebra relying on the Hlder inequality. We give an appealing alternative proof relying on elementary convex analysis that we hope is pedagogically useful. The core lemma is the following. Let K ⊂ Rⁿ be a convex cone and g: K → R_≥0 be a positively homogeneous function with g(x) > 0 for x ≠ 0. Then, g is convex (resp. concave) if and only if the sublevel set {x ∈ K: g(x) ≤ 1} (resp. its complement) is convex. This yields a nice characterization of a norm via its unit ball. As roots and powers preserve the sublevel set at height 1, another immediate consequence is the following: if f: K → R_≥0 is a convex (resp. concave) positively homogeneous function of degree p ≥ 1 (resp. 0 < p ≤ 1), with f(x) > 0 for x ≠ 0, then g(x):= [f(x)]^1/p is convex (resp. concave). This readily implies the Minkowski and reverse Minkowski inequalities; also some other applications are briefly exemplified.