  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.
