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A basic theorem of projective geometry states that any bijection that sends real projective $n$-space to itself and takes projective lines to projective lines is a projective transformation (it is induced from an invertible linear map from $\mathbb{R}^{n+1}$ to itself).

Question. Is a bijection (or a homeomorphism) from an open convex subset of $\mathbb{R}^n$ to itself that sends line segments to line segments the restriction of a projective transformation?

I think this is true (and must be known) if the open convex subset is the ball (or an ellipsoid).

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    $\begingroup$ Take a look at the phd thesis (and related papers) of Rupert Mccallum rupertmccallum.com $\endgroup$
    – Uri Bader
    Commented Aug 6, 2016 at 13:32
  • $\begingroup$ I asked a related question a while ago. Sergei Ivanov's answer there may be relevant. $\endgroup$
    – LSpice
    Commented Aug 6, 2016 at 13:48
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    $\begingroup$ One more comment about the ball case that you mentioned: this incidence geometry is usually called "the Klein model of the hyperbolic space" and indeed, the result is well known in this case (as well as in other cases, if you follow my link above). $\endgroup$
    – Uri Bader
    Commented Aug 6, 2016 at 14:10

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