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shadow12345

convex vs manifold

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A triangle is convex, but when I think convex, I think of a 3D object that is enclosed but also has a non zero volume, such as a circular cylinder. Is my definition for convex really manifold, or is manifold something different?

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Convex is defined in any number of dimensions >= 2. A set of points (such as the set of points within a triangle) is convex if no two points within the set form a line segment which passes outside the set.

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Quote:
Original post by shadow12345Is my definition for convex really manifold, or is manifold something different?

Convex is, as Sneftel pointed out, a set of points such that any segment whose ends are in the set is completely contained in the set. The number of dimensions doesn't need to be >=2, though; a segment is a perfectly good convex set.

A manifold is an object that looks like R^n locally everywhere. For instance, a sphere looks locally like R^2 (if you only look a little bit around you, the Earth looks flat, like a plane). You can get a more precise definition by googling.

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