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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 $f_0(x) \exp(\rho(x))$ with a given density $f_0$ and $\rho$ 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. This is joint work with Peter McCullagh, Alexandre Moesching and Christof Straehl.