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

Local continuity of log-concave projection, with applications to estimation under model misspecification