  We show a general phenomenon of the constrained functional value for
densities satisfying general convexity conditions, which generalizes the
observation in Bobkov and Madiman (2011) that the entropy per coordinate in a
log-concave random vector in any dimension with given density at the mode has a
range of just 1. Specifically, for general functions $\phi$ and $\psi$, we
derive upper and lower bounds of density functionals taking the form $I_\phi(f)
= \int_{\mathbb{R}^n} \phi(f(x))dx$ assuming the convexity of $\psi^{-1}(f(x))$
for the density, and establish the tightness of these bounds under mild
conditions satisfied by most examples. We apply this result to the distributed
simulation of continuous random variables, and establish an upper bound of the
exact common information for $\beta$-concave joint densities, which is a
generalization of the log-concave densities in Li and El Gamal (2017).
