A nonabelian Brunn–Minkowski inequality

• Published in: Geometric and functional analysis Vol. 33; no. 4; pp. 1048 - 1100
• Main Authors: Jing, Yifan, Tran, Chieu-Minh, Zhang, Ruixiang
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-08-2023; Springer Nature B.V
• Subjects: Analysis; Group theory; Lie groups; Linear algebra; Mathematics; Mathematics and Statistics; Subgroups; Primary 22D05; 05D99; Secondary 43A05; 60B15; 49Q20
• ISSN: 1016-443X; 1420-8970
• DOI: 10.1007/s00039-023-00647-6

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.