  We continue the study undertaken in \cite{GHN} of left democracy function
$h_l(N)=\inf_{#\Lambda=N}\left\|\sum_{n\in \Lambda_N} x_n\right\| $ of an
unconditional basis in a Banach space $X$. We provide an example of a basis
with $h_l$ non-doubling. Then we show that for bases with non-doubling $h_l$
the greedy projection is not optimal. Together with results from \cite{GHN} and
improved by C. Cabrelli, G. Garrig\'os, E. Hernandez and U. Molter we get that
the basis is greedy if and only if the greedy projection is optimal.
