Let $S,T \subset \mathbb{R}^n$ be measurable sets, and suppose that there exists a measurable bijection $f\colon S\to T$ so that $$ \f(x)f(y)\ \;\geq\; \xy\ $$ for all $x,y \in S$. Does it follow that $\mu(S) \leq \mu(T)$?
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It follows from two observations:


