Talk:Hadwiger's theorem
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Can someone please clarify the point concerning the Euler characteristic? By self-similarity, the measure of a cube in R3 must be zero, but one can easily construct a finite union of cubes which is homotopy equivalent to the 2-sphere, and the latter has non-zero Euler characteristic by all definitions I know.--80.133.118.97 12:30, 22 Mar 2005 (UTC)
- I think what you're missing is that the condition that the measure is additive does not mean that it is additive over union, but rather over disjoint union (think about it:
, but 
). So the idea of building objects in 3-space using cubes as bricks simply does not work, as the disjoint union of two cubes is disconnected (conversely, since Rn is Hausdorff, any nontrivial finite decomposition of a compact connected set in Rn yields at least one non-compact subset; a cube minus one of its faces, if you like). --86.130.247.29 22:59, 25 July 2005 (UTC)
- I see, thank you.--80.136.146.247 18:12, 10 August 2005 (UTC)
Is it correct to replace "not-necessarily-nonnegative set functions" by the more straightforward phrasing "real-valued functions" or am I missing something (which is quite likely as I know nothing about this subject)?
Crust 16:32, 30 August 2005 (UTC)
Why isn't the number of connected components a second example (after Euler characteristic) of a homogenous measure of degree 0 (contrary to the asserted uniqueness)? (Maybe the theorem is only supposed to refer to disjoint unions not general unions of finitely many convex sets. If so, the Euler characteristic is just the number of connected components in this case.)
Crust 16:50, 31 August 2005 (UTC)
I've split the first paragraph into two, so that the new first para contains a cleaner statement of the theorem and the new second para explains what "measure" means in this context. I added to the definition of "measure" the requirements that it be rotation-invariant and continuous, since these are both hypotheses in Hadwiger's Theorem. I removed the reference to scalar multiplication, since this is already included within the term "linear combination".
130.209.6.40 16:36, 30 January 2007 (UTC) (Tom Leinster)
