  How should an agent (the sender) observing multi-dimensional data (the state
vector) persuade another agent to take the desired action? We show that it is
always optimal for the sender to perform a (non-linear) dimension reduction by
projecting the state vector onto a lower-dimensional object that we call the
"optimal information manifold." We characterize geometric properties of this
manifold and link them to the sender's preferences. Optimal policy splits
information into "good" and "bad" components. When the sender's marginal
utility is linear, revealing the full magnitude of good information is always
optimal. In contrast, with concave marginal utility, optimal information design
conceals the extreme realizations of good information and only reveals its
direction (sign). We illustrate these effects by explicitly solving several
multi-dimensional Bayesian persuasion problems.
