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I've been trying to analyse Steiner symmetrization. I've asked before about the symmetrization preserving $\textit{volume}$, I'm still going over the same proof. I think I understand a bit what is going on.

I have one issue however, the slides I'm going through state that:

$\int_{x_n \in G_x \cap \Omega}dx_n = m(x_1,...,x_{n-1},0) = \int_{x_n \in G_x \cap S(\Omega)}dx_n$

Can someone please take a peek at the slides and explain if you can.

I think I understand why $\int_{x_n \in G_x \cap \Omega}dx_n = m(x_1,...,x_{n-1},0)$. It is sayin' that the result is just function of the other variables without $x_n$, the variable we were integrating with respect to.

What I do agree with is that $\int_{x_n \in G_x \cap \Omega}dx_n = m_1(x_1,...,x_{n-1},0)$ and $\int_{x_n \in G_x \cap S(\Omega)}dx_n = m_2(x_1,...,x_{n-1},0)$. Why is $m_2(x_1,...,x_{n-1},0) = m_1(x_1,...,x_{n-1},0)$ though?

Community
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Adeeb
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1 Answers1

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It's probably better to think over what they're saying. For each value of $(x_1,\ldots,x_{n-1})$, the Steiner symmetrization is the map that sends the slice of $\Omega$ with those values to the interval of the same length as the slice, centered about the origin.

All your integral says is that the measure of the slice is the measure of the same slice on the Steiner-symmmetrized set, which is just true by the way the symmetrization was defined.

Ray Yang
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