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Consider $\mathbb{R}^n$ with the Hausdorff metric, $$d(A,B) = \max(\sup_{a\in A}\inf_{b \in B} ||a-b||,\sup_{b\in B} \inf_{a\in A}||a-b||).$$ I'm looking for a reference containing a statement like the following: $\lim_{i\to 0} A_i = A$ if and only if $A$ is the set of all limits of convergent sequences $\{x_i\}$ with $x_i\in A_i$.

My interest comes from reading Introduction to Tropical Geometry by Diane Maclagan and Bernd Sturmfels (AMS, latest edition 2021). Chapter Three is Tropical Varieties, and in Sec. 3.6 (Stable Intersection) there is a proposition concerning the stable intersection of two weighted balanced polyhedral complexes in $\mathbb R^n$.

The prefactory remarks to that result say the Hausdorff metric "lets us speak about the limit of a sequence of subsets of $\mathbb R^n$."

If the subsets are weighted polyhedral complexes $\Sigma_i$ that converge to a polyhedral complex $\Sigma$, then the limit inherits a weighting in the following way. A top-dimension cell $\sigma$ of the limit complex $\Sigma$ is the limit of top-dimensional cells $\sigma_i$ of $\Sigma_i$ if $\lim_{i\to \infty} \sigma_i = \sigma$. We consider the set of all such sequences $\sigma_i$ limiting to $\sigma$, where we identify cofinal sequences. If $\lim_{i\to \infty} \operatorname{mult}_{\Sigma_i} (\sigma_i)$ exists for all such sequences, then we define the multiplicity of $\sigma$ to be the sum of all these limits.

The case of finite collections of weighted points is then mentioned by way of illustration: "the multiplicity of a limit point $\mathbf u$ is then the sum of multiplicities of all points that tend to $\mathbf u$." Prop. 3.6.12 is then a result "that works in general."

Since tropical varieties in $\mathbb R^n$ are not necessarily compact, some accomodation is needed to apply the Hausdorff metric to a notion of a convergent sequence of polyhedral complexes. The quoted passage suggests that this is in part accomplished by breaking up consideration of the polyhedral complex into its top-dimensional cells.

I'd appreciate a reference in which the details of that limiting construction are spelled out.

hardmath
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ThLD
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  • Hi @hardmath. Thanks for your quick reply. Maybe I was imprecise. It is not the actual proof but more a reference to a book containing the statement, that I am searching for. – ThLD May 08 '22 at 16:12
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    For various book references, see my comment to Finding example of metric space where you can determine the structure of a metric space with hausdorff metric, especially Chapter III -- specifically pp. 121-124 and some of the exercises on p. 135 -- in the freely available 1969 book Point Sets by Eduard Čech. – Dave L. Renfro May 08 '22 at 16:58
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    This is simply false as stated. – Moishe Kohan May 09 '22 at 02:15
  • Some points to clarify. Presumably $A,B$ are subsets of $\mathbb R^n$. Are they compact? What does $\lim_{i\to 0} A_i$ mean? You haven't defined $A_i$ nor a sense in which the index $i$ tends to zero. Does $i$ have a bearing on how $A_i$ is derived from $A$? – hardmath May 09 '22 at 03:33
  • @hardmath; you are hitting the head on the nail! These are exactly the questions I have myself. I'm looking at a proof, where the reader is assumed to know about the Hausdorff metric. Thus there are lots of missing/implicit details - for example whether the sets are compact. So I am trying to backtrack to the original statement/theory. Asking for a reference was my try to make the question short and precise. However, that might not have been the best way to do it... – ThLD May 09 '22 at 08:45
  • @hardmath: Of course, I'm looking at Prop. 3.6.12 in Introduction to Tropical Geometry by Maclagan and Sturmfels. See [here] (https://books.google.dk/books?id=c6ZaEAAAQBAJ&lpg=PA147&ots=IyM5AcrsBV&dq=tropical%20geometry%20proposition%203.6.12&hl=da&pg=PA142#v=onepage&q=tropical%20geometry%20proposition%203.6.12&f=false). The sets $\Sigma_1 \cap (\epsilon v+\Sigma_2)$ are closed, but as far as I have understood, they are not necessarily bounded. However, this seems to be crucial when defining the Hausdorff metric. – ThLD May 09 '22 at 18:30
  • @hardmath Sure :-) – ThLD May 09 '22 at 18:48
  • Have a look over my edit, and rollback or amend if your meaning is unintentionally changed. There are certainly many technical details that are omitted in the vicinity of the proposition you cited, but I suspect they are in most cases (e.g. definition of stable interesection) filled in by earlier parts of the book. Note that besides $\lim_{i\to \infty}$ there is also in the passage $\lim_{\epsilon \to 0}$, which we should be careful to distinguish. – hardmath May 10 '22 at 18:59
  • @hardmath: Thanks a lot for your help! – ThLD May 11 '22 at 11:51

1 Answers1

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Your limit as $i \to 0$ should be as $i \to \infty$. See Theorem $2.10$ here. It describes the limit of a Cauchy sequence in the space $H(X)$ of all nonempty closed and bounded subsets of a complete metric space $X$, where $H(X)$ is equipped with the Hausdorff metric. Theorem $2.20$ shows the space $K(X)$ of nonempty compact subsets of $X$ is a closed subset of $H(X)$ if (and only if) $X$ is complete.

You are interested in the case $X = \mathbf R^n$, when $H(X) = K(X)$.

KCd
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  • Hi @KCd. Thanks a lot for your answer and the link to the very well-written notes! It seems like the boundedness condition is crucial for the Hausdorff distance to be a metric. However, the statement I'm looking at concerns unbounded polyhedral complexes (see edit of OP). Do you know if there is an (easy) way to extend the Hausdorff metric to (a class of) unbounded set? – ThLD May 14 '22 at 19:05
  • That is not anything I have experience with, sorry. In $\mathbf R^2$, for the sets $A = {(x,y) : xy = 1}$ and $B = {(x,y) : xy = 0}$ (a rectangular hyperbola and the $x$ and $y$ axes), the supremum $\sup_{a \in A} d(a,B)$ is infinite, so for the Hausdorff distance to make sense on whatever unbounded sets you want to use, some kind of extra condition must be needed for the distances to be finite. – KCd May 14 '22 at 19:41
  • Alright, thanks for your help! – ThLD May 16 '22 at 08:20