The question
I am just exploring different properties of Steiner symmetrization and under continuities I have conjectured that for a given function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ that is:
- non-negative,
- supported on a ball, and
- of finite $L_1$ norm (for simplicity let's just say its range is $\{0,1\}$),
there exists a constant $c>0$ such that for all $p,q\in\mathbb{R}P^{n-1}$ the inequality $$||S_p(f)-S_q(f)||_1\leq c\cdot d(p,q),$$ where $S_p$ denotes the Steiner symmetrization operator along the direction $p$ and $d$ denotes some "natural" metric on $\mathbb{R}P^{n-1}$ like the angle.
a little story
I know and understand all the properties of Steiner symmetrization that are found on its Wikipedia page or cognates of inequalities for the symmetric decreasing rearrangement that are also found on its Wikipedia page (except the Polya-Szego inequality), but this one is not in that list yet seems so true and natural. Is it true?
My progress
I have proved that this is true if we not only fix $f$, but $p$ too. But honestly I think the full question is much harder. Also, I have some very convincing ideas for why $f$ has to be fixed. At best, $c$, as a function of $f$, could be continuous. But that's only my guesses.