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Let $f:{\mathbb R}\rightarrow{\mathbb R}_+$ be a log-convex function. Suppose that $f_{\epsilon}$ is the smoothed version of $f$:

$$f_{\epsilon}(x)=\int \varphi_{\epsilon}(x-y)f(y)dy,$$

where $\varphi$ is a Mollifier. I would appreciate any pointers on relevant literature (or a hint) referring to conditions on $\varphi$ such that $f_{\epsilon}$ is log-convex.

Thanks.

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Yes if you assume $\phi_\epsilon\ge0$. Because $f_\epsilon$ belongs to the convex cone spanned by the shifted functions $f(\cdot-h)$. Since every $f(\cdot-h)$ is logarithmically convex and the set of logarithmically convex functions is itself a convex cone, you are done.

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  • $\begingroup$ Thanks. I would also like to know what happens if $f$ depends on other variables,e.g. a parametric function, and is log-convex with respect to those variables (the parameters) that are not being Mollified. $\endgroup$
    – Nigel
    Commented Jan 8, 2015 at 15:28
  • $\begingroup$ Very nice answer for the question! $\endgroup$
    – Changyu Guo
    Commented Jan 12, 2015 at 20:57

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