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I am tinking that probably Andrew Stacey or somebody can give me an immediate answer to the following question, without me having to think about it:
take X a convenient vector space , let Δn be the standard n-simplex regarded as a smooth manifold and consider the mapping vector space XΔn.
I want to consider the collection C∞(XΔn,ℝ) of all smooth functions on XΔn, and the collection C∞(LnXΔ•) of all “functions on the subspace of degenerate simplices” and ask if the restriction map
C∞(XΔn)→C∞(LnXΔ•)is surjective.
Here that collection of functions on degenerate simplices is the evident limit over function sets C∞(XΔk) for k<n.
In other words: given an function on the set of all degenerate smooth n-simplices Δn→X that is smooth when restricted to a function on k-simplices for all k<n, does it always have an extension to a smooth function on the space of all n-simplices?
My guess would be that it does extend, and that an extension is given by something like
f(x0,…,xn)=(−1)n∑(−1)|I|fI(xI)The signs may be wrong, of course.
The simplest case is considering the positive quadrant in the xy-plane and two functions fx,fy:[0,∞)→ℝ with fx(0)=fy(0). Then define f:[0,∞)×[0,∞)→ℝ by f(x,y)=fx(x)+fy(y)−f0(0), where f0:{0}→ℝ is f0(0)=fx(0).
Thanks. Let’s see, you are thinking of maps that embed the simplex with straight edges and faces, right? so that only the images of the vertices need to be specified? I was thinkig of all smooth maps Δn→X.
Should have made it clearer: the I in the summation is a multi-index. So you sum over all proper subsets of {0,…,n}. The idea being an extension of the example given with two variables.
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