[Paper Review] A nonabelian Brunn-Minkowski inequality

Yifan Jing, Chieu-Minh Tran+1 more|arXiv (Cornell University)|Jan 19, 2021

Point processes and geometric inequalities13 references4 citations

TL;DR

This paper establishes a nonabelian Brunn-Minkowski inequality for locally compact groups, proving it is sharp for helix-free groups including real linear algebraic groups, semisimple and solvable Lie groups. The proof uses induction on dimension, introduces a modified inequality for nonunimodular groups, and employs a proportionated averaging technique to generalize the classical Brunn-Minkowski inequality beyond abelian settings.

ABSTRACT

Henstock and Macbeath asked in 1953 whether the Brunn-Minkowski inequality can be generalized to nonabelian locally compact groups; questions along the same line were also asked by Hrushovski, McCrudden, and Tao. We obtain here such an inequality and prove that it is sharp for helix-free locally compact groups, which includes real linear algebraic groups, Nash groups, semisimple Lie groups with finite center, solvable Lie groups, etc. The proof follows an induction on dimension strategy; new ingredients include an understanding of the role played by maximal compact subgroups of Lie groups, a necessary modified form of the inequality which is also applicable to nonunimodular locally compact groups, and a proportionated averaging trick.

Motivation & Objective

• To generalize the classical Brunn-Minkowski inequality to nonabelian locally compact groups.
• To resolve a long-standing open problem posed by Henstock, Macbeath, Hrushovski, McCrudden, and Tao regarding nonabelian generalizations.
• To establish a sharp inequality that applies to unimodular and nonunimodular groups, including real linear algebraic and Lie groups.
• To develop a framework that accounts for the modular function and non-invariant Haar measures in nonunimodular settings.

Proposed method

• Employing induction on the dimension of the group to reduce the problem to simpler substructures.
• Introducing a modified Brunn-Minkowski-type inequality that incorporates both left and right Haar measures to handle nonunimodular groups.
• Using a proportionated averaging trick to control measure growth across group products.
• Analyzing the role of maximal compact subgroups in Lie groups to decompose the structure and control measure behavior.
• Reducing the problem to cocompact and codiscrete subgroups via short exact sequences.
• Applying the Gleason–Yamabe theorem and structure theory of almost-Lie groups to handle general locally compact groups.

Experimental results

Research questions

• RQ1 Can the Brunn-Minkowski inequality be generalized to nonabelian locally compact groups, particularly in the nonunimodular case?
• RQ2 What is the correct form of the inequality when the Haar measure is not right-invariant?
• RQ3 For which classes of locally compact groups is the inequality sharp?
• RQ4 How does the dimension of maximal compact subgroups influence the measure growth in group products?
• RQ5 What structural properties of Lie groups (e.g., solvability, semisimplicity) affect the validity and sharpness of the inequality?

Key findings

• The paper establishes a sharp nonabelian Brunn-Minkowski inequality for helix-free locally compact groups, including real linear algebraic groups and semisimple Lie groups with finite center.
• The inequality is shown to be sharp for unimodular solvable Lie groups of dimension d with maximal compact subgroup of dimension m, where μ(XY)^{1/(d−m)} ≥ μ(X)^{1/(d−m)} + μ(Y)^{1/(d−m)} holds.
• For nonunimodular groups, a modified inequality is introduced that involves both left and right Haar measures, ensuring meaningful lower bounds on μ(XY).
• The proof reveals that nonunimodular cases are essential for the inductive argument, even when focusing on unimodular groups.
• The proportionated averaging technique successfully controls measure distortion in non-invariant settings, enabling the generalization.
• The result confirms that McCrudden’s inequality for solvable Lie groups is sharp and extends it to broader classes of groups.