On Minkowski symmetrizations of $\alpha$-concave functions and related applications

A Minkowski symmetral of an $\alpha$-concave function is introduced, and some of its fundamental properties are derived. It is shown that for a given $\alpha$-concave function, there exists a sequence of Minkowski symmetrizations that hypo-converges to its ``hypo-symmetrization". As an application, it is shown that the hypo-symmetrization of a log-concave function $f$ is always harder to approximate than $f$ is by ``inner log-linearizations" with a fixed number of break points. This is a functional analogue of the classical geometric result which states that among all convex bodies of a given mean width, a Euclidean ball is hardest to approximate by inscribed polytopes with a fixed number of vertices. Finally, a general extremal property of the hypo-symmetrization is deduced, which includes a Urysohn-type inequality and the aforementioned approximation result as special cases.