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I want to find a visual proof of the following fact:

For any convex figure in the plane there is a sequence of Steiner's symmetrizations that makes it arbitrary close to a circular disc.

All proofs I know require some integral estimates. I would prefer a more visual proof (even if it is more involved).

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  • $\begingroup$ Do you consider Blaschke selection theorem visual? $\endgroup$ – Ivan Izmestiev Aug 1 '18 at 15:07
  • $\begingroup$ @IvanIzmestiev, no I want a direct proof. (I guess you want to say that optimal shape exists and it must be round disc and therefore everything converges to the disc --- no I do not want it.) $\endgroup$ – Anton Petrunin Aug 1 '18 at 15:16
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(1) Let $$ C = {\rm conv}\ B\bigcup \{x\}$$ where $B$ is a closed ball of center $o$ and $x$ is not in $B$.

Consider arc $A\subset B$, $$ \partial B \bigcap {\rm Int}\ C $$

Note that by any Steiner symmetrization $S_L$, $A$ is still in the interior ${\rm Int}\ S_L( C)$.

If $x_1$ is close to $x$ s.t. $\overrightarrow{ox}$ and $\overrightarrow{ox_1}$ are different, then $S_L(C)$ where $L=( \overrightarrow{ox_1})^\perp$ has arc $A_1$ which contains $A$. After a finite process, $\partial B-A_n$ is a one point set. Hence through a some Steiner symmetrization, $A_{n+1}=\partial B$. Hence the resulting convex and compact set contains a ball $B'$ whose radius is strictly larger than that of $B$.

(2) Fix a point $o\in \mathbb{E}^2$ which is the origin. Here assume that all hyperplane pass through $o$.

Assume that $C$ is a convex and compact subset. If $x$ is interior point in $C$, then define $v=\overrightarrow{ox}$ so that $o$ is an interior point of $S_L(C)$ for $v^\perp=L$.

In further, if $B$ is a largest closed ball in $C$ of center $o$, then there is an open arc of non-zero length in $ \partial B \bigcap {\rm Int}\ C$.

Hence we have $B'$ in $(1)$. So repeat the process.

If there is no such arc in $\partial B \bigcap {\rm Int}\ C$, then $B=C$.

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Just for illustration, here is an animation of one step of a Steiner symmetrization applied to a rasterized version of a polygon.


                    SSpoly


          SSk1s6
          First: a histogram of vertical lengths. Second: lengths lifted to be symmetric around horizontal.


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    $\begingroup$ Thanks for the visualization. I think it would be a nice addition to the Wikipedia article on Steiner symmetrization :) $\endgroup$ – Shamisen Aug 6 '18 at 15:37
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    $\begingroup$ @Shamisen: Yes, I was thinking of that myself. Requires a block of time to master exactly how to incorporate a new image into a Wikipedia article. On my ToDo list... $\endgroup$ – Joseph O'Rourke Aug 6 '18 at 16:04

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