Convexity in the real projective plane?

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In summary, the conversation discusses the use of "convex sets" in an affine patch of the real projective plane RP^2. The speaker clarifies that using an affine patch should not be a problem as manifolds have a local Euclidean feature. The other person agrees and thanks them for their response.
  • #1
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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  • #2
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.
 
  • #3
ok, that's what I though too. Thanks!
 

1. What is Convexity in the real projective plane?

Convexity in the real projective plane refers to the property of a set or shape where any line segment connecting two points within the set remains entirely within the set. In simpler terms, it is a measure of how "curved" or "bent" a shape is.

2. How is Convexity different from convexity in Euclidean geometry?

Convexity in the real projective plane differs from convexity in Euclidean geometry in that it takes into account the concept of infinity. In Euclidean geometry, a shape is considered convex if any line segment connecting two points within the shape lies entirely within the shape. In the real projective plane, this also applies to points at infinity, which are points that have no specific location but are considered to be infinitely far away from the origin.

3. What is the importance of Convexity in the real projective plane?

Convexity in the real projective plane is a fundamental concept in mathematics and has several applications in fields such as computer graphics, robotics, and data analysis. It helps in understanding the structure and properties of shapes and is also used in optimization problems to find the most efficient solutions.

4. How is Convexity measured in the real projective plane?

Convexity is measured using the concept of convex hulls in the real projective plane. A convex hull is the smallest convex shape that contains a given set of points. The more "curved" a shape is, the larger its convex hull will be, and the less convex it is considered to be.

5. Can a shape be convex in Euclidean geometry but not in the real projective plane?

Yes, a shape can be convex in Euclidean geometry but not in the real projective plane. This is because the concept of convexity in the real projective plane takes into account points at infinity, which are not considered in Euclidean geometry. Therefore, a shape that is convex in Euclidean geometry may not be convex in the real projective plane if it has points at infinity that lie outside the shape's convex hull.

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