  We study an information design problem with continuous state and discrete
signal space. Under convex value functions, the optimal information structure
is interval-partitional and exhibits a dual expectations property: each induced
signal is the conditional mean (taken under the prior density) of each
interval; and each interval cutoff is the conditional mean (taken under the
value function curvature) of the interval formed by neighboring signals. This
property enables an examination into which part of the state space is more
finely partitioned. The analysis can be extended to general value functions and
adapted to study coarse mechanism design.
