https://en.wikipedia.org/wiki/Steinhaus_theorem This is the theorem I'm referring to, the wiki only states it if the dimension is 1. Is it true in higher dimensions?
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Yes, the theorem holds for any lebesgue measurable subset of $\mathbb{R}^n$. See here – Friedrich Apr 30 '20 at 22:51
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There is a proof using convolution of functions. That proof works for $\mathbb R^{n}$. – Kavi Rama Murthy Apr 30 '20 at 23:20
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The proof in your Wiki link is valid, verbatim, for $\Bbb R^n$ for any finite $n$. – DanielWainfleet May 01 '20 at 00:01