By $\mathcal{S}$ let us denote the set of such step functions $f:[0,1]\to [0,1]$ that additionally satisfy: $$\forall_{ x>\frac{1}{2}} \ \ \lambda\Big(f=x\Big) \ = \ x\cdot \Big[\lambda\Big(f=x\Big)+\lambda\Big(f=1-x\Big)\Big],$$ and $$\forall_{ x>\frac{1}{2}} \ \ \lambda\Big(f=1-x\Big) \ = \ (1-x)\cdot \Big[\lambda\Big(f=x\Big)+\lambda\Big(f=1-x\Big)\Big],$$ where $\lambda$ is simply the Lebesgue measure. This means that measure designated to the set $\{x,1-x\}$ is split proportionaly between the points $x$ and $1-x$.
I am interested in evaluating (finding or bounding) $$\sup_{f \in \mathcal{S}} \ \int\limits_{0}^{1}\Big|f^i-f^d\Big|^p \ \mathrm{d}\lambda, $$ where $f^i$ and $f^d$ are increasing and decreasing rearrangements of $f$ and $p>1$:
My questions are:
- Have You seen similar problems? What is Your guess, how could one deal with such wired optimization?
- I suspect that the number of values attained by the best $f$ is equal $3$ $-$ namely $(1, x, 1-x)$ for some $x$. Do You know any methods that could be helpful in proving that the number of values of $f$ under the consideration could be bounded by some value? I was thinking about Caratheodory theorem from convex geometry, but I don't see any way of applying it.
Thank You for any insight.