Martingale Type, the Gamlen-Gaudet Construction and a Greedy Algorithm

In the present paper we identify those filtered probability spaces $(\mathrm{\Omega },\mathcal{F},\left({\mathcal{F}}_n\right),\mathbb{P})$ that determine already the martingale type of a Banach space $X$. We isolate intrinsic conditions on the filtration $({\mathcal{F}}_n)$ of purely atomic $\sigma$-algebras which determine that the upper ${\mathrm{\ell }}^p$ estimates $$\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p(\mathrm{\Omega },X)}^p\le C^p(\mathrm{\parallel }\mathbb{E}f\mathrm{\mid }{\mathcal{F}}_0{\mathrm{\parallel }}_{L^p(\mathrm{\Omega },X)}^p+\sum _{n=1}^{\mathrm{\infty }}\mathrm{\parallel }{\mathrm{\Delta }}_{n}f{\mathrm{\parallel }}_{L^p(\mathrm{\Omega },X)}^p),f\in L^p(\mathrm{\Omega },X)$$ imply that the Banach space $X$ is of martingale type $p$. Our paper complements \mbox{G. Pisier’s} investigation \cite{Pisier1975} and continues the work by S. Geiss and second named author in \cite{Geiss2008}.