We show that a (semi-normalized) basis in a Banach space is quasi-greedy with quasi-greedy constant equal to 1 if and only if it is unconditional with suppression-unconditional constant equal to 1.
@article{arxiv.1504.04368,
title = {Characterization of 1-quasi-greedy bases},
author = {Fernando Albiac and Jose L. Ansorena},
journal= {arXiv preprint arXiv:1504.04368},
year = {2015}
}