2023/01/30 by Hoehner, Steven
#52A40 (Secondary) #52A41 (Primary) 39B62 #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2301.12619
A Minkowski symmetral of an α-concave function is introduced, and some of its fundamental properties are derived. It is shown that for a given α-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.