  For a locally compact group $G$, let ${\rm AM}({\rm A}(G))$ denote the
amenability constant of the Fourier algebra of $G$. We show that ${\rm AM}({\rm
A}(G))\geq 3/2$ for every non-abelian $G$, extending a result of Johnson (JLMS,
1994) who obtained this for finite non-abelian groups. This lower bound, which
is known to be best possible, improves on results by previous authors and
answers a question raised by Runde (PAMS, 2006).
  Our approach uses a minorant for ${\rm AM}({\rm A}(G))$, related to the
antidiagonal in $G\times G$, which was implicitly used in Runde's work but
which has not been studied systematically before. Our main novelty is an
explicit formula for this minorant when $G$ is a countable virtually abelian
group, in terms of the Plancherel measure for $G$. We also establish some
general properties of this minorant, and present some examples to support the
conjecture that the minorant coincides with the amenability constant.
