  In this manuscript, we derive optimal conditions for building function
approximations that minimize variance when used as importance sampling
estimators for Monte Carlo integration problems. Particularly, we study the
problem of finding the optimal projection $g$ of an integrand $f$ onto certain
classes of piecewise constant functions, in order to minimize the variance of
the unbiased importance sampling estimator $E_g[f/g]$, as well as the related
problem of finding optimal mixture weights to approximate and importance sample
a target mixture distribution $f = \sum_i \alpha_i f_i$ with components $f_i$
in a family $\mathcal{F}$, through a corresponding mixture of importance
sampling densities $g_i$ that are only approximately proportional to $f_i$. We
further show that in both cases the optimal projection is different from the
commonly used $\ell_1$ projection, and provide an intuitive explanation for the
difference.
