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Has anyone thought about approximating smoothly embedded balls in $\mathbb{R}^3$ by polyhedra, and taking the limit of Dehn invariants to obtain a "smooth Dehn invariant"? I am guessing that you would have to be careful about allowable approximations to get a well defined invariant.

Just curious if there have been any developments along these lines.

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Dehn invariants are valuations so, from this point, of view there has been enormous progress on this topic (see the work of Klain, Alesker, and Ludwig, to name three distict lines of research). However, Dehn invariants are not continuous in the Hausdorff topology and it may be hard to control those "allowable" approximations. –  alvarezpaiva Mar 21 '13 at 19:48
    
I would guess that you would need to capture geometry, not just topology. I was thinking something like normal vectors to the boundary faces converging to normal vectors to the surface. –  Steve Mar 21 '13 at 22:54
    
The only valuation I know that captures topology is the Euler characteristics, the usual ones (mixed volumes for example) measure various geometric averages. –  alvarezpaiva Mar 23 '13 at 13:36
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