Convexity in the real projective plane?

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SUMMARY

The discussion centers on the application of the concept of a "convex set" within an affine patch of the real projective plane, denoted as RP^2. Participants confirm that utilizing convex sets in this context is valid, as affine patches resemble Euclidean spaces locally. The consensus is that working within a single chart allows one to disregard the non-Euclidean nature of RP^2, making the use of convex sets appropriate.

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klabautermann
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Hi,

can I use the notion of a "convex set" in an affine patch of the real projective plane RP^2, or is there any danger in doing that? With affine patch I mean any sybset of RP^2 that does not include points at infinity w.r.t some coordinate chart, e.g., RP^2 \ {(x:y:0)|x,y real}.

Thanks,
klabautermann
 
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I don't think this should be a problem. The central feature of manifolds is that they look locally like a Euclidean space. If you can do everything inside a single chart, then you might as well forget completely that you're in a non-Euclidean space.
 
ok, that's what I though too. Thanks!
 

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