TY - JOUR

T1 - A proof of the Minkowski inequalities based on convex homogeneous functions

AU - Karrila, Seppo

AU - Karrila, Taewee

AU - Karrila, Alex

N1 - Funding Information:
We would like to thank the referees for their comments and suggestions on the manuscript. A.K. wishes to thank Mikael Lindström for discussions and the Academy of Finland (grant #339515) for financial support.
Publisher Copyright:
© 2020 by TJM. All rights reserved.

PY - 2022

Y1 - 2022

N2 - 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.

AB - 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.

KW - convexity of real functions of several variables

KW - functional analysis

KW - functional inequalities

UR - http://www.scopus.com/inward/record.url?scp=85140375637&partnerID=8YFLogxK

UR - http://thaijmath.in.cmu.ac.th/index.php/thaijmath/article/view/6108

M3 - Article

AN - SCOPUS:85140375637

SN - 1686-0209

VL - 2022

SP - 98

EP - 105

JO - Thai Journal of Mathematics

JF - Thai Journal of Mathematics

IS - Special Issue

ER -