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Rupert
Posts:
3,797
Registered:
12/6/04
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ping José Carlos Santos
Posted:
Dec 21, 2012 3:28 AM
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Previously I made the assertion to you that if f is a mapping from a
Euclidean space E^n into a Euclidean space E^m which maps straight
lines into straight lines and whose range has three points in general
position, then it follows that f is an affine transformation.
As Guowu Yao of Tsinghau University has pointed out to me this is
false. For consider the case n=m=2 and f acts as the identity on a
straight line and collapses the rest of the plane to a point not on
the line.
However I believe that I can fix this. For example, I think that I can
prove that if f is a mapping from a Euclidean space E^n into E^n,
which maps straight lines into straight lines, and whose range
contains n+2 distinct points any n+1 of which are in general position,
then f is an affine transformation.
I hope to publish this sometime soon. I will keep you posted on the
details.
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