My question is as follows:
a) Let M be a C^r manifold in R^m of dimension k (assume say k < m)
b) Let us consider the set R_+M = { rz : r >= 0, z in M } equally as a subset of R^m ;
c) Can we conclude that R_+M is a C^r manifold too, and if so how is the dimension impacted. (I am guessing it is not larger than k + 1). If true how would one show this, or what reference to a (positive or negative) result could you provide?
conic sections are well defined for 1,2,3 and 4 dimensions... i do not see what would prevent the generalization beyond. the intersection of a hypersurface with an Rn hyper-cone should generalize the conic section.
Dear Prof Klatchko,
I am not sure to understand your reply. I am interested in the conic hull generated by the manifold. Not so much the other way around.
Best Regards
Wim
Let we сonsider a non trivial knot in R^3. Cone on it will not be a manifold in R^3.
The origin will be a problematic point. For example, in the plane, consider an open, bounded segment and the cone over it. You will end up with a vertex at the origin, where you cannot define a tangent plane.
If for each z in M, the half-line {rz : r>0} intersects M only at the point z, then you will have a (k+1)-dimensional manifold whose boundary will be M.
Locally, you can construct a tubular neighbourhood around each point of M. With it, you can prove that the cone is going to be a manifold at most of the points, since you might find some problematic points in the boundary of the cone.
Dear Miguel,
Thank you for your reply. In my case, 0 does not belong to M ; In particular one can find an open ball around 0 that does not intersect M at all.
Does this then mean that you have positively answered the question ? Do you have a reference to recommend?
Best Regards
Wim
About 0:Consider a circle in a 2-plane in R^3, far away from the origin. Your cone over this circle will be the famous quadric in R^3 called "the cone". But at 0, the cone it's not a manifold.
Sorry, I don't have any reference. Anyway, there is a huge literature on convex geometry.
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