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Hadwiger's theorem of integral geometry states that all continuous valuations which are invariant under rigid motions are expressible in terms of the intrinsic volumes. The continuity property means with respect to the Hausdorff distance: $$d_H(X, Y) = \max{( \sup_{x \in X} \inf_{y \in Y} d(x,y), \sup_{y \in Y} \inf_{x \in X} d(x,y) )}$$ where $X,Y \subseteq \mathbb{R}^n$ and $d(\cdot,\cdot)$ is the Euclidean distance metric.

Now, the zero dimensional intrinsic volume is the Euler characteristic $\chi$. I am confused by how $\chi$ is continuous with respect to the Hausdorff distance. Consider the following example: two identical balls $A,B$ having diameter $\sigma$ are separated by a distance $r$. The Euler characteristic of their union is a valuation, i.e. $$\chi(A \cup B) = \chi(A) + \chi(B) - \chi(A \cap B)$$ And according to standard texts on integral geometry (e.g. Klain & Rota Introduction to Geometric Probability (2006)) this is continuous with respect to the distance metric above. However, explicitly the Euler characteristic is discontinuous: $$\chi(A \cup B) = \begin{cases}1 & \forall \; r < 2\sigma \\ 2 & \forall \; r > 2\sigma\end{cases}$$ whereas the Hausdorff distance between them is simply their separation $d_H(A,B)\equiv r$.

How is the Euler characteristic in my counter example continuous with respect to $d_H(A,B)$?

Edit: It is quite rightly pointed out in the comments that Hadwiger’s theorem applies to strictly convex sets. I am in fact assuming a second extension theorem due to Groemer which generalises the result to so-called polyconvex sets, i.e. sets formed by countable union of convex sets.

Arctic Char
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Joshua Robinson
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    I do not know this area, but the linked statement says that Hadwiger's theorem is true for convex bodies, not unions thereof. –  Dec 08 '18 at 19:27
  • The theorem is generalisable to polyconvex sets, meaning any body formed by countable union of convex sets. My example is the union of two convex sets, where the discontinuity occurs as they become disjoint. – Joshua Robinson Dec 08 '18 at 22:10
  • Which is why I am skeptical of your first sentence. ;) You are of course completely correct that there is a jump in the Euler characteristic of this continuously varying family. There must be something about the hypotheses that goes wrong here. –  Dec 08 '18 at 22:16
  • Agreed, I have misunderstood something; the question is what. I will dig up the generalisation of the theorem and update my question. – Joshua Robinson Dec 08 '18 at 22:31
  • Could you give a textbook reference (page/theorem number) for the precise statement of this generalization (or possibly the original paper)? I will gladly take a look. –  Dec 09 '18 at 01:19
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    Certainly. In Klain & Rota (see post above for full reference) the relevant theorems are: Groemer’s extensions theorem 5.1.1 on page 44, the volume characterisation theorem 8.1.5 on page 105 and Hadwiger’s theorem 9.1.1 on page 118. – Joshua Robinson Dec 09 '18 at 10:22

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Klain and Rota only claim (and only prove) that Euler characteristic is convex-continuous, meaning it's only continuous with respect to sequences of convex compact subsets. This is slightly buried in the discussion in Chapter 4 because they use "continuous" to name this condition in this chapter, which I personally think is very confusing (it took me the last half hour or so to notice this).

But the theorem establishing the existence of the Euler characteristic (Theorem 5.2.1) is careful to specify "convex-continuous," as is the statement of Hadwiger's theorem (Theorem 9.1.1). And Groemer's extension theorem (Theorem 5.1.1) only claims that a convex-continuous valuation extends to a valuation on polyconvex sets, not that the resulting valuation is continuous on polyconvex sets.

If you don't want to restrict to convex sets, I think the correct the sense in which the Euler characteristic is continuous is that it's continuous on the space of step functions spanned by indicator functions of polyconvex sets (it's a measure against which such step functions can be integrated). What happens when two balls (or more general sets) collide is that the corresponding sum of indicator functions is equal to $2$ on their intersection. So the integral against the Euler characteristic has an extra term corresponding to the Euler characteristic of the intersection, and for balls it remains $2$ as expected.

Qiaochu Yuan
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