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[Edited Jan 23, 2021]

Let $D^n$ be the $n$-dimensional unit radius disk in euclidean $\mathbb{R}^n$.

Larry Guth's Sponge Problem asks: Does there exist a constant $\epsilon=\epsilon_n$ such that every open subset $U\subset \mathbb{R}^n$ satisfying $vol(U)< \epsilon_n$ admits an expanding embedding $f: U\hookrightarrow D^n$?

Recall $f$ is expanding embedding (EE) iff $f$ is a continuous embedding and the symmetric matrix ${}^tDf\cdot Df$ has all eigenvalues $\geq 1$. This last local condition holds iff $||D_xf(v)||\geq ||v||$ for every $x\in U$ and tangent vector $v\in T_x U$. Equivalently $f$ is (EE) iff $f$ increases the induced path length of all curves in $U$, where the path length is defined by continuous paths contained in the image $f(U)$.

My previous best estimate $\epsilon_n^*$ for $\epsilon_n$ comes from obvious example of two disks $D_1, D_2$ of radius $1/2$ kissing, which disks $U':=D_1\cup D_2$ cannot be properly expanded embedded into $D$. (The two kissing disks barely embed into $D$, even though they occupy only $1/2^{n-1}$ a fraction of the volume of $D^n$). Similar examples arise from "truncated" Apollonian packings, e.g. the first figures of Sarnak's MAA address http://web.math.princeton.edu/sarnak/InternalApollonianPackings09.pdf

Possible counterexamples arise among low density disk packings which are strongly jammed, as investigated by S. Torquato and F.H. Stillinger, e.g. consider the following figures which are reproduced from Figure 2, https://arxiv.org/abs/cond-mat/0112319 . In the figure 2 enter image description here

we see a packing which is rigidly jammed, and which cannot be anywhere locally expanded embedded into a smaller disk (or rectangle in this case). The construction of such rigidly jammed packings which have arbitrarily small density is described in https://www.degruyter.com/view/journals/zkri/220/7/article-p657.xml For example, consider the figure 2 in Fischer, reproduced below enter image description here The rigidity of these packings is a form of incompressibility which can possibly lead to counterexamples to the Sponge Problem.

We emphasize that the packings in the above figures cannot be compressed (contained within a smaller boundary) via a global expanding-embedding. For example, there are no volume preserving rigid motions which preserve tangencies between all the disks, and which compresses the packing into a smaller volume.

Question: Can anybody provide further update on the status of L.Guth's Sponge Problem? Are better approximations $\epsilon^*$ known, or candidate open sets $U\subset \mathbb{R}^n$ with $vol(U)<\epsilon_n^*$ which cannot be expanded embedded into $D$?

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    $\begingroup$ 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 $D$. $\endgroup$
    – JHM
    Jan 14, 2021 at 11:22
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    $\begingroup$ 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. $\endgroup$ Jan 14, 2021 at 11:42
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    $\begingroup$ 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 . $\endgroup$
    – JHM
    Jan 14, 2021 at 12:11
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    $\begingroup$ @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!) $\endgroup$ Jan 14, 2021 at 13:04
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    $\begingroup$ 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. $\endgroup$ Jan 18, 2021 at 7:48

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If one removes the injectivity condition, then the answer is "yes". (At least in the two-dimensional case.) That is, there is a lengh-increasding immersion $U\looparrowright D$.

Indeed, note that $U$ can be sliced by line segments from-boundary-to-boundary into subsets of diameter $100\cdot\sqrt{\varepsilon}$. Arrange these pieces near the center of $D$. Connect the corresponding cuts by immersed road in $D$ of the same width. Finally, note that we can spread a neighborhood of each cut along the road in a length-increasing way.

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  • $\begingroup$ Preferably any proof of Sponge Problem would make no appeals to width-volume inequalities (since Guth's motivation for Sponge Problem was to find new independant proof of Gromov's width-volume inequalities). I should have made this clear in the question. Do you know of elementary proof that $U$ can be partitioned into relative cycles of diameter $<100 \sqrt{\epsilon}$ (or $100^{100^{100}}\sqrt{\epsilon}$) in dimension two? $\endgroup$
    – JHM
    Jan 20, 2021 at 13:36
  • $\begingroup$ @IHM it follows from Besicovitch, is not it? $\endgroup$ Jan 20, 2021 at 18:47
  • $\begingroup$ If it follows from Besicovitch (eg. arxiv.org/pdf/2010.10040) then i dont see how, especially if $U$ is an open subset of small volume (possibly countably infinite genus in two dimensions). $\endgroup$
    – JHM
    Jan 21, 2021 at 1:58
  • $\begingroup$ @JHM you can always assume that it is bounded by a finite number of smooth closed curves. $\endgroup$ Jan 21, 2021 at 4:41
  • $\begingroup$ The above immersion is strictly locally length increasing, or am I mistaken? (where distance on the image is induced path length as a subset of $D$). My definition of expanding-embedding is incomplete: it needs hypothesis that $f$ is injective. And is it clear that your map is continuous? Namely, when you "arrange these pieces" and when you "connect the corresponding cuts..." how to ensure a continuous map is being defined? It's interesting that the volume of the image $f(U)\subset D$ can perhaps(!) be made arbitrarily small by your construction, an "expanding collapse". $\endgroup$
    – JHM
    Jan 23, 2021 at 16:28

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