In the lecture note http://www.math.utoronto.ca/almut/rearrange.pdf, it was stated that the volume of the set of critical points decreases under symmetric decreasing rearrangement. It seems so obvious however I cannot prove it. There is also deleted topic in mathoverflow which stated exactly the same question as mine. Can someone help me? Thanks in advance!
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$\begingroup$ well, a critical point is a change from (nonstrict) increasing to (nonstrict) decreasing, or vice versa, right? But a monotone function gets rid of the nonstrictness. You are left with only strict changes. $\endgroup$– Ben WCommented Oct 4, 2018 at 0:07
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