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 hl non-doubling. Then we show that for bases with non-doubling hl 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.
Cite
@article{arxiv.1303.4972,
title = {On left democracy function},
author = {P. Wojtaszczyk},
journal= {arXiv preprint arXiv:1303.4972},
year = {2013}
}