In this talk we discuss nonparametric estimation of a log-concave density and nonparametric estimation of a tail inflation function (McCullagh and Polson, 2012) in a common framework. A variation of the latter model assumes data coming from a density f0(x)exp(ρ(x)) with a given density f0 and ρ being an unknown convex function, a so-called tail inflation function. We discuss and illustrate a variation of the active set approach by Duembgen, Huesler and Rufibach (2007/2011) which works in both settings.