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The list of known Banach spaces whose linear geometry determines the(nonlinear) democracy functions of their quasi-greedy bases to the extent thatthey end up being democratic, reduces to $c_0$, $\ell_2$, and all separable$\mathcal{L}_1$-spaces. Oddly enough, these are the only Banach spaces that,when they have an unconditional basis, it is unique. Our aim in this paper isto study the connection between quasi-greediness and democracy of bases innonlocally convex spaces. We prove that all quasi-greedy bases in $\ell_p$ for$0<p<1$ (which also has a unique unconditional basis) are democratic withfundamental function of the same order as $(m^{1/p})_{m=1}^\infty$. The methodswe develop allow us to obtain even more, namely that the same occurs in anyseparable $\mathcal{L}_p$-space, $0<p<1$, with the bounded approximationproperty.

Quasi-greedy bases in the spaces $\ell_p$ ($0