Image may be NSFW.
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I just came back from a splendid visit to Singapore and Vietnam and I will write about it later. While I was away, Nathan Rubin organized a lovely conference on topics closed to my heart ERC Workshop: Geometric Transversals and Epsilon-Nets with many interesting lectures. Nathan himself announced a great result with new upper bounds for planar weak ε-nets. In 2007 I wrote a guest post (my first ever blog post) on the topic on Terry Tao’s blog. The next section is taken from that old post with reference to one additional result from 2008.
Weak ε-Nets
Definition: Let Image may be NSFW.
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be a finite set of points, and let Image may be NSFW.
Clik here to view.. We say that a finite set Image may be NSFW.
Clik here to view. is a weak Image may be NSFW.
Clik here to view.-net for X (with respect to convex bodies) if, whenever B is a convex body which is large in the sense that Image may be NSFW.
Clik here to view., then B contains at least one point of Y. (If Y is contained in X, we say that Y is a strong Image may be NSFW.
Clik here to view.-net for X with respect to convex bodies.)
let Image may be NSFW.
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be the least number such that every finite set X possesses at least one weak Image may be NSFW.
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-net for X with respect to convex bodies of cardinality at most Image may be NSFW.
Clik here to view.. (One can also replace the finite set X with an arbitrary probability measure; the two formulations are equivalent.) Informally, f is the least number of “guards” one needs to place to prevent a convex body from covering more than Image may be NSFW.
Clik here to view. of any given territory.
A central problem in discrete geometry is:
Problem: For fixed d, what is the correct rate of growth of f as Image may be NSFW.
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?
It is already non-trivial (and somewhat surprising) that Image may be NSFW.
Clik here to view. is even finite. This fact was first shown by Alon, Bárány, Füredi, and Kleitman (the planar case was achieved earlier by Bárány, Füredi, and Lovász), who established a bound of the form Image may be NSFW.
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; this was later improved to Image may be NSFW.
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by Chazelle, Edelsbrunner, Grigni, Guibas, Sharir, and Welzl. In the other direction, the best lower bound was for many years Image may be NSFW.
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for some positive c(d); it was shown by Matousek that c(d) grows at least as fast as Image may be NSFW.
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for some absolute constant c. In 2008 Bukh, Matousek, and Nivasch proved that Image may be NSFW.
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.
Nathan Rubin’s new theorem
In the plane the state of our knowledge was that (up to constants) for every Image may be NSFW.
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) Image may be NSFW.
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. The new dramatic breakthrough is
Theorem (Nathan Rubin, 2018): Image may be NSFW.
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.
I hope the paper will be arxived in the next few months.
Other results from Ein-Gedi
Here is a page with the titles and abstracts of the lectures. A lot of very interesting advancements on Helly-type theorems, transversals, geometric-Ramsey, topological methods and other topics. Let me briefly mention one direction.
Improved (p,q)-theorems
Weak Image may be NSFW.
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-nets play an important role in the proof by Alon and Kleitman of the Hadwiger-Debrunner conjecture. The proof is so nice that I could not resist the temptation to present and discuss it in two comments (1,2) to my old 2007 posts.
Theorem (Alon and Kleitman): For every integers Image may be NSFW.
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and Image may be NSFW.
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, Image may be NSFW.
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, there is a function Image may be NSFW.
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such that the following is true:
Every finite family Image may be NSFW.
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of convex sets in Image may be NSFW.
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with the property that among every Image may be NSFW.
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sets in the family there are Image may be NSFW.
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with non empty intersection can be divided to at most Image may be NSFW.
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Dramatic improvements of the bounds for Image may be NSFW.
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Other results around weak ε-Nets
There were various other developments regarding weak ε-Nets andrelated notions that occured since my 2007 post and the result of Buck, Matousek, and Nivasch in 2008, and probably I am not aware of all of them (or even forgot some). Of course, comments about related results are most welcome in the comment section. Here are links to a lecture by Noga Alon: 48 minutes on strong and weak epsilon nets (Part 1, Part 2). Let me mention just two beautiful recent developments: A paper On weak ε-nets and the Radon number by Moran and Yehudayoff gives a simple proof for their finiteness in a very abstract setting. A paper by Har-Peled and Jones How to Net a Convex Shape studies interesting weaker notions. A weaker notion of weak epsilon nets is described in the work by Bukh and Nivash in Nivash’s Ein Gedi abstract.