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user9072
user9072
Typo in title. Added paragraph breaks.
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edit approved Jan 19, 2016 at 21:29
jeq
edit approved Jan 19, 2016 at 21:29
jeq
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Two questions about the extent to which simple arcs and simple closed curves can fill up higher dimrnsionaldimensional Euclidean spaces

For each positive integer n, let E(n) be n-dimensional Euclidean space with its standard metric and let p(n) be some fixed point of E(n). The so-called "Osgood Curve" shows that there can exist simple arcs and simple closed curves in E(2) which have positive 2-dimensional Lebesgue measure. 

Question (1): Do there exist-for arbitrarily large n-simple arcs or simple closed curves in E(n) which have positive n-dimensional Lebesgue measure? 

Question (2): Is it true-for arbitrarily large n-that there exist simple arcs or simple closed curves in E(n) which do not contain p(n) but which intersect each ray in E(n) originating from p(n) in at least one point? This is clearly true when n=2.

Two questions about the extent to which simple arcs and simple closed curves can fill up higher dimrnsional Euclidean spaces

For each positive integer n, let E(n) be n-dimensional Euclidean space with its standard metric and let p(n) be some fixed point of E(n). The so-called "Osgood Curve" shows that there can exist simple arcs and simple closed curves in E(2) which have positive 2-dimensional Lebesgue measure. Question (1): Do there exist-for arbitrarily large n-simple arcs or simple closed curves in E(n) which have positive n-dimensional Lebesgue measure? Question (2): Is it true-for arbitrarily large n-that there exist simple arcs or simple closed curves in E(n) which do not contain p(n) but which intersect each ray in E(n) originating from p(n) in at least one point? This is clearly true when n=2.

Two questions about the extent to which simple arcs and simple closed curves can fill up higher dimensional Euclidean spaces

For each positive integer n, let E(n) be n-dimensional Euclidean space with its standard metric and let p(n) be some fixed point of E(n). The so-called "Osgood Curve" shows that there can exist simple arcs and simple closed curves in E(2) which have positive 2-dimensional Lebesgue measure. 

Question (1): Do there exist-for arbitrarily large n-simple arcs or simple closed curves in E(n) which have positive n-dimensional Lebesgue measure? 

Question (2): Is it true-for arbitrarily large n-that there exist simple arcs or simple closed curves in E(n) which do not contain p(n) but which intersect each ray in E(n) originating from p(n) in at least one point? This is clearly true when n=2.

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Garabed Gulbenkian
Garabed Gulbenkian
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Two questions about the extent to which simple arcs and simple closed curves can fill up higher dimrnsional Euclidean spaces

For each positive integer n, let E(n) be n-dimensional Euclidean space with its standard metric and let p(n) be some fixed point of E(n). The so-called "Osgood Curve" shows that there can exist simple arcs and simple closed curves in E(2) which have positive 2-dimensional Lebesgue measure. Question (1): Do there exist-for arbitrarily large n-simple arcs or simple closed curves in E(n) which have positive n-dimensional Lebesgue measure? Question (2): Is it true-for arbitrarily large n-that there exist simple arcs or simple closed curves in E(n) which do not contain p(n) but which intersect each ray in E(n) originating from p(n) in at least one point? This is clearly true when n=2.