  The main aim of this work is to study important local Banach space constants
for Boolean cube function spaces. Specifically, we focus on
$\mathcal{B}_{\mathcal{S}}^N$, the finite-dimensional Banach space of all
real-valued functions defined on the $N$-dimensional Boolean cube $\{-1,
+1\}^N$ that have Fourier--Walsh expansions supported on a fixed~family
$\mathcal{S}$ of subsets of $\{1, \ldots, N\}$. Our investigation centers on
the projection, Sidon and Gordon--Lewis constants of this function space. We
combine tools from different areas to derive exact formulas and asymptotic
estimates of these parameters for special types of families $\mathcal{S}$
depending on the dimension $N$ of the Boolean cube and other complexity
characteristics of the support set $\mathcal{S}$. Using local Banach space
theory, we establish the intimate relationship among these three important
constants.
