Calculation of symmetric-decreasing rearrangement

Question

I can't image the symmetric-decreasing rearrangement, so , I want to calculate some example, but fail. For example,how to calculate the symmetric-decreasing rearrangement of $f(x)=x$ on $[0,10]$, zero elsewhere?

Answer 1

Suppose $f(x) = \chi_{[0,10]}(x) x.$ First step: $$ \{ x \colon f(x) > t\} = \{ x \colon \chi_{[0,10]}(x) x > t\} = (t \wedge 10, 10] $$ for $t \ge 0.$ Second step: $$ |(t \wedge 10, 10]| = 10 - (t \wedge 10) = (10-t)^+. $$ Third step: $$ \{x \mid f(x) > t\}^* = \Big\{y \colon |y| < \frac{(10-t)^+}{2}\Big\} $$ Final step: \begin{align*} f^*(x) &= \int_0^\infty \chi_{\{y \colon f(y) > t\}^*}(x) dt\\ &= \int_0^\infty \chi_{\{y \colon |y| < (10-t)^+/2\}}(x) dt\\ &= \int_0^{10} \chi_{\{|x| < (10-t)/2\}} dt\\ &= \int_0^{10} \chi_{\{t < 10 - 2|x|\}} dt\\ &= \int_0^{(10-2|x|)^+}dt\\ &= (10-2|x|)^+\\ &= \chi_{[-5,5]}(x) (10-2|x|). \end{align*}