Let E(n) be n-dimensional Euclidean space. It is known that there exist subsets of E(n) which are simple arcs and have positive n-dimensional Lebesgue measure when n=1 or 2. Does this continue to be true for arbitrarily large n? If not, what is the largest n for which it holds and is there a simple proof of this fact? Intuitively, I feel that there should be no upper bound, but cannot see how to prove it.
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asked May 6, 2013 at 17:48
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$\begingroup$ Yes (this is so for every $n=1\ 2\ \ldots$). $\endgroup$– Włodzimierz HolsztyńskiCommented May 7, 2013 at 0:04
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$\begingroup$ Thanks alot for your answer. I have searched extensively to find it in the literature but with no success. $\endgroup$– Garabed GulbenkianCommented May 7, 2013 at 19:42
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4$\begingroup$ mathoverflow.net/questions/228819/… $\endgroup$– Douglas ZareCommented Jan 24, 2016 at 17:07
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