The Brunn-Minkowski inequality

2002/04/08 by Richard J. Gardner · 6 citations
Mathematics · #Point processes and geometric inequalities #Mathematics and Applications #Mathematical Inequalities and Applications #Isoperimetric inequality #Inequality #Minkowski inequality #Mathematics #Minkowski space #Kantorovich inequality #Pure mathematics #Mathematical economics #Rearrangement inequality #Log sum inequality #Mathematical analysis #Linear inequality #Geometry

paper · pdf · doi:10.1090/s0273-0979-02-00941-2

openalex publication_date 2002/04/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The Brunn-Minkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of \mathbb Rⁿ, and deserves to be better known. This guide explains the relationship between the Brunn-Minkowski inequality and other inequalities in geometry and analysis, and some applications.

Citations

• Ellipsoids of maximal volume in convex bodies
• Best constants in Young's inequality, its converse, and its generalization to more than three functions
• A Mathematical Theory of Communication
• Inequalities in Fourier Analysis
• Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality
• Analysis
• Logarithmic Sobolev Inequalities

Cited by

• Optimal Transport
• Log-concavity of asymptotic multigraded Hilbert series
• A unified treatment for Lp Brunn-Minkowski type inequalities
• A local proof of the dimensional Prékopa's theorem
• Convex body [wikipedia]
• John ellipsoid [wikipedia]