  We establish a Shearer-type inequality for the Poincar\'e constant, showing
that the Poincar\'e constant corresponding to the convolution of a collection
of measures can be nontrivially controlled by the Poincar\'e constants
corresponding to convolutions of subsets of measures. This implies, for
example, that the Poincar\'e constant is non-increasing along the central limit
theorem. We also establish a dimension-free stability estimate for
subadditivity of the Poincar\'e constant on convolutions which uniformly
improves an earlier one-dimensional estimate of a similar nature by Johnson
(2004). As a byproduct of our arguments, we find that the monotone properties
of entropy, Fisher information and the Poincar\'e constant along the CLT find a
common root in Shearer's inequality.
