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Lately I've been reading about Hausdorff distance and the Fell topology, which are both essentially topologies on subsets of $2^X\setminus \{ \emptyset \}$. The use I saw of this notion is of discussing the limit of spectra of operators, but I'm assuming there are plenty of uses which I don't suspect to look for.

Can anyone please give some examples of intersting Applications\uses of Hausdorff distance and such 'hyper'-topologies? Even with relation to simplifying other known mathametical problems.

Keen-ameteur
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    There are several hundred papers (at least) in which the Baire category theorem is applied to hyperspaces of sets to either easily obtain the existence of very pathological sets, or simply to show that "most" compact sets have certain properties. Examples of the former can be found with this search and an example of the latter is the fact that "most" compact subsets of ${\mathbb R}^n$ are perfect, nowhere dense, and have Lebesgue measure zero (indeed, Hausdorff dimension $0$ and even smaller). – Dave L. Renfro Oct 23 '20 at 10:57
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    Those spaces (and Effros-Borel spaces) are also important in descriptive set theory for a variety of reasons, whole books have been written on them, personally I like the one by Nadler (it's called Hyperspaces something but I can never remember the whole name) – Alessandro Codenotti Oct 23 '20 at 13:41
  • @Alessandro Codenotti: Curiously, I recently ordered 7 or 8 books from amazon about a week ago (I do this 2 to 4 times a year, depending on how much I can afford to spend) and one of them is Nadler's book Hyperspaces of Sets, a book I've seen and looked at in libraries since it came out in 1978 but have never gotten around to buying a copy (which wasn't easy to do until the past 20 years or so, when online buying made locating copies possible). (continued) – Dave L. Renfro Oct 23 '20 at 15:48
  • Two related books, each of which I've known about since they showed up in libraries after publication but which I still haven't gotten around to obtaining copies of, are Topologies on Closed and Closed Convex Sets by Gerald Alan Beer (1993) and Hausdorff Approximations by Blagovest Hristov Sendov (1990). – Dave L. Renfro Oct 23 '20 at 15:54
  • For some references (very concisely provided) regarding the size of "most" compact sets in the sense of Baire category (in the space of compact sets under the Hausdorff metric), see my comments to The collection of all compact perfect subsets is $G_\delta$ in the hyperspace of all compact subsets. In my last comment at this earlier question, the MR citations are the Mathematical Reviews codes of the papers, and the page numbers given are the page to look at in the paper itself. – Dave L. Renfro Oct 23 '20 at 16:00
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    On the off-chance that anyone is interested, I've since gotten (hardback) copies of Beer's book and Sendov's book, as well as Nadler's Continuum Theory (1992). I should have gotten a good used copy of the last one, as Marcel Dekker's lowering of book binding standards makes a new copy worse than a reasonably good early 1990s used version. I've found this to be the case with Springer-Verlag texts also . . . – Dave L. Renfro Apr 26 '22 at 17:45

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One of practical applications of Hausdorff distance is image comparison. For instance, the first ten entries for a respective Google query provided me the following references.

Riadh Bouslimi, Jalel Akaichi, Using Hausdorff Distance for New Medical Image Annotation, International Journal of Database Management Systems 4:1 (Feb. 2012).

Normand Grégoire, Mikael Bouillot, Hausdorff distance between convex polygons, Web project presented to Mr. Godfried Toussaint for the course CS 507 Computational Geometry McGill University, fall 1998.

Daniel P. Huttenlocher, Gregory A. Klanderman, William J. Rucklidge, Comparing Images Using the Hausdorff Distance, IEEE transactions of pattern analysis and machine intelligence 15:98 (Sept. 1993) 850–863.

Oliver Jesorsky, Klaus J. Kirchberg, Robert W. Frischholz, Robust Face Detection Using the Hausdorff Distance, in: Proc. Third International Conference on Audio- and Video-based Biometric Person Authentication, Halmstad, Sweden, 6–8 June 2001, Springer, Lecture Notes in Computer Science, LNCS-2091, 90–95.

K. Senthil Kumar, T. Manigandan, D. Chitra, L. Murali, Object recognition using Hausdorff distance for multimedia applications, Multimedia Tools and Applications 79 (2020) 4099–4114.

Xiao Hong Li, Yi Zhen Jia, Feng Wang, Yuan Chen Image Matching Algorithm Based on an Improved Hausdorff Distance, 2nd International Symposium on Computer, Communication, Control and Automation (3CA 2013), 244–247.

Barnabás Takács, Comparing face images using the modified Hausdorff distance, Pattern Recognition 31:12 (Dec. 1998) 1873–1881.

Chyuan-Huei Thomas Yang, Shang-Hong Lai, Long-Wen Chang, Hybrid image matching combining Hausdorff distance with normalized gradient matching, Pattern Recognition 40 (2007) 1173–1181.

Alex Ravsky
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