Timeline for Status of Larry Guth's Sponge Problem
Current License: CC BY-SA 4.0
22 events
| when toggle format | what | by | license | comment | |
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| Dec 28, 2021 at 21:28 | comment | added | JHM | I think the stable packings constructed by Werner Fischer (see the above figures) are effectively counter examples to Guth's Sponge Problem. Stable packings do not have any strictly local deformations -- if any sphere is moved, it is moved by a global deformation. And this prevents any expanding embedding from deforming the packing into a smaller domain. | |
| Feb 23, 2021 at 16:00 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Jan 24, 2021 at 15:28 | history | edited | JHM | CC BY-SA 4.0 |
Clarifying the question.
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| Jan 23, 2021 at 17:16 | history | edited | JHM | CC BY-SA 4.0 |
added images and references. clarified error w.r.t. Apollonian packings.
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| Jan 20, 2021 at 5:02 | comment | added | Anton Petrunin | @JHM, the total area of the roads might be arbitrary small. | |
| Jan 19, 2021 at 22:34 | answer | added | Anton Petrunin | timeline score: 1 | |
| Jan 19, 2021 at 12:38 | comment | added | JHM | @ArseniyAkopyan No, I don't know how. But would such an open set be any different from the disk? (If I connected them by roads, my roads would fill the disk). Do you have an image/picture of what your roads would look like? | |
| Jan 18, 2021 at 7:48 | comment | added | Arseniy Akopyan | Do you know how to map the following set? Take all rational points on a circle and connect every pair of them by a thin road, in a such way that total area is small. | |
| Jan 18, 2021 at 7:46 | comment | added | Arseniy Akopyan | @JHM dropbox.com/s/p5fthz04z81z8hs/dorozinski2006.pdf?dl=0 | |
| Jan 15, 2021 at 12:24 | comment | added | JHM | Reportedly there exist "rigid/jammed" packings of arbitrarily low density, but the articles are behind degruyter paywall. degruyter.com/view/journals/zkri/221/5-7/…. If these novel packings N are *incompressible" then maybe $\epsilon^*$ is zero. | |
| Jan 14, 2021 at 21:25 | history | edited | JHM | CC BY-SA 4.0 |
corrected notation.
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| Jan 14, 2021 at 21:05 | history | edited | JHM | CC BY-SA 4.0 |
edited. added @Balarka observations.
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| Jan 14, 2021 at 20:10 | comment | added | JHM | @BalarkaSen Yes!, i think your comment contains a nice improvement of $\epsilon^*$. If we replace any open set $U$ with an Apollonian packing $A$ of itself of minimal volume, then $vol(A)<vol(U)$ and $U$ e-embeds into $D$ iff $A$ e-embeds into $D$. So replacing the two kissing disks with an apollonian packing of minimal volume would improve $\epsilon^*$ by some factor. | |
| Jan 14, 2021 at 13:04 | comment | added | Balarka Sen | @JHM I was thinking of the following, does this work? Take your example, $D_1 \cup D_2$ where each disk has radius $r_1 = r_2 = 1/2 + \varepsilon$. Now drill a smaller disk $D_3 \subset D_2$ out, and fill the hole $D_3$ in by an Apollonian disk packing by open disks. The pores are so densely populated inside $D_3$ that they don't seem "squishable", and this has clearly less area than your example $D_1 \cup D_2$. Is this the true sponge idea? (Edit: Thanks for those references, by the way!) | |
| Jan 14, 2021 at 12:11 | comment | added | JHM | Annuli appear to be no different than rectangles (with respect to e-embeddings), but I don't have a precise criterion for which rectangles e-embed into $D$ (except volume and disjoint disks). All this sponge stuff appears to have begun with Ya. Barzdin and A. Kolmogorov. On realization of nets in 3-dimensional space. Problems of Cybernetics, 19:261–268, 1967. But I have not studied that paper. Some introduction can be found in last chapter of P.G.Adey's thesis pgadey.com/ut-thesis.pdf . | |
| Jan 14, 2021 at 11:43 | history | edited | JHM | CC BY-SA 4.0 |
added 58 characters in body
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| Jan 14, 2021 at 11:42 | comment | added | Balarka Sen | Thanks for clarifying! Do you know what could be done with annuli? It seems easy to see that an annuli around a circle of radius $r$ of thickness $C(r)$ expanded embeds in $D(0; 1)$ only if $C(r) = O(1/r)$ (one needs to make it very wrinkly), but I'd have to think about the constants. | |
| Jan 14, 2021 at 11:22 | comment | added | JHM | If reference disk $D$ has radius $r=1$, then the kissing disks have radius $r_1=r_2=1/2$. The interiors of the kissing disks are disjoint, and can be embedded into $D$, but every arbitrarily small $\epsilon$-thickening of $D_1 \cup D_2$, or pair of kissing disks of radius $r_1=r_2=1/2+\epsilon$ cannot be expanded embedded into $D$. | |
| Jan 14, 2021 at 10:44 | history | edited | JHM | CC BY-SA 4.0 |
deleted 1 character in body
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| Jan 14, 2021 at 7:25 | comment | added | Balarka Sen | Do you mean disks of diameter 1/2 + eps for any eps > 0 (which does yield your bound)? Else for the two kissing disks of diameter 1/2 each, which is contained in D(0, 1), you can bulge parts of one out staying inside D(0, 1), which gives an expanding embedding. Am I missing something? | |
| Jan 14, 2021 at 3:54 | comment | added | katago | What is the best upper bound of $\epsilon_n^{*}$? | |
| Jan 14, 2021 at 2:38 | history | asked | JHM | CC BY-SA 4.0 |