Date: Dec 23, 2012 1:40 PM
Author: Jose Carlos Santos
Subject: Re: ping JosÃ© Carlos Santos
On 21/12/2012 08:28, Rupert wrote:

> 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.

Thank you. I'll be waiting for your proof.

Best regards,

Jose Carlos Santos