An extremal property of the symmetric decreasing rearrangement

Let $\alpha$ be a given real number. It is shown that for a given $\alpha$-concave function, its symmetric decreasing rearrangement is always harder to approximate in the symmetric difference metric by $\alpha$-affine minorants with a fixed number of break points. This extends a classical result of Macbeath (1951) from convex bodies to a functional setting.
Comment: 11 pages, 1 figure