  The log-concave projection is an operator that maps a d-dimensional
distribution P to an approximating log-concave density. Prior work by
D{\"u}mbgen et al. (2011) establishes that, with suitable metrics on the
underlying spaces, this projection is continuous, but not uniformly continuous.
In this work we prove a local uniform continuity result for log-concave
projection -- in particular, establishing that this map is locally
H{\"o}lder-(1/4) continuous. A matching lower bound verifies that this exponent
cannot be improved. We also examine the implications of this continuity result
for the empirical setting -- given a sample drawn from a distribution P, we
bound the squared Hellinger distance between the log-concave projection of the
empirical distribution of the sample, and the log-concave projection of P. In
particular, this yields interesting statistical results for the misspecified
setting, where P is not itself log-concave.
