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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 However I think the least dense nonembeddable disk packing of $D$ is the two kissing disks.

Possible counterexamples arise among low density disk packings which are strongly jammed, as occurs in research of S. Torquato and F.H. Stillinger, e.g. consider the figures taken from Figure 2, https://arxiv.org/abs/cond-mat/0112319 . In the figure 2

we see a packing which is rigidly jammed, and which cannot be anywhere locally expanded embedded into a smaller disk (here rectangle). And 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 The rigidity of these packings is a form of incompressibility which can possibly lead to counterexamples to the Sponge Problem.

Question: Is anybody aware of any further progress in Guth's Sponge Problem, or candidate open sets $U\subset \mathbb{R}^n$ with $vol(U)<\epsilon_n^*$ which cannot be expanded embedded into $D$?

[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

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 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 $D$?

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 iff the symmetric matrix ${}^tDf\cdot Df$ has all eigenvalues $\geq 1$. Equivalently iff $||D_xf(v)||\geq ||v||$ for every $x\in U$ and tangent vector $v\in T_x U$. Equivalently iff $f$ increases the length of all curves in $U$.

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 $D$). Remarks of @Balarka Sen in comments below yields the following slightly volume decreasing improvement. Let $A$ be an Apollonian packing of $U'$ which has minimal volume. (Recall $A$ is some infinite maximal disk packing of $U'$). Then we estimate $$\epsilon^*_n=\inf_{A} vol(A),$$ where the infimum is taken over all Apollonian packings $A$ of $U'$.

Obviously $\epsilon_n^*$ is slightly smaller than the volume of $U'$, which is $1/2^{n-1}$ times the volume of the unit disk. Replacing $U'$ by an Apollonian packing $A$ is interesting idea, but does not dramatically reduce the volume.

Question: Is anybody aware of any further progress in Guth's Sponge Problem, or candidate open sets $U\subset \mathbb{R}^n$ with $vol(U)<\epsilon_n^*$ which cannot be expanded embedded into $D$?

[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 However I think the least dense nonembeddable disk packing of $D$ is the two kissing disks.

Possible counterexamples arise among low density disk packings which are strongly jammed, as occurs in research of S. Torquato and F.H. Stillinger, e.g. consider the figures taken from Figure 2, https://arxiv.org/abs/cond-mat/0112319 . In the figure 2

we see a packing which is rigidly jammed, and which cannot be anywhere locally expanded embedded into a smaller disk (here rectangle). And 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 The rigidity of these packings is a form of incompressibility which can possibly lead to counterexamples to the Sponge Problem.

Question: Is anybody aware of any further progress in Guth's Sponge Problem, or candidate open sets $U\subset \mathbb{R}^n$ with $vol(U)<\epsilon_n^*$ which cannot be expanded embedded into $D$?

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 iff the symmetric matrix ${}^tDf\cdot Df$ has all eigenvalues $\geq 1$. Equivalently iff $||D_xf(v)||\geq ||v||$ for every $x\in U$ and tangent vector $v\in T_x U$. Equivalently iff $f$ increases the length of all curves in $U$.

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 $D$).

The remarks of @Balarka Sen in comments yield the following improvement. Let $A$ be an Apollonian packing of $U$ which has minimal volume. Then we estimate $\epsilon^*_n$ as the minimal volume of an Apollonian packing of two kissing disks of radius $1/2$. This yields $$\epsilon^*_n<2 \cdot \text{Vol}(D(0,1/2))=\frac{\pi^{n/2}}{2^{n-1} \cdot\Gamma(n/2+1)},$$ where $D(0,1/2)$ is the $n$-ball of radius $1/2$.

I've been interested in Sponge problem for a few years, and for all that time I've yet to improve on the estimate $\epsilon^*_n = \epsilon_n$. (Although replacing $U$ with an Apollonian packing $A$ of minimal volume is new idea I learned from the comments).

Question: Is anybody aware of any progress in Guth's Sponge Problem, or candidate open sets $U\subset \mathbb{R}^n$ with $vol(U)<\epsilon_n^*$ which cannot be expanded embedded into $D$?

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 iff the symmetric matrix ${}^tDf\cdot Df$ has all eigenvalues $\geq 1$. Equivalently iff $||D_xf(v)||\geq ||v||$ for every $x\in U$ and tangent vector $v\in T_x U$. Equivalently iff $f$ increases the length of all curves in $U$.

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 $D$). Remarks of @Balarka Sen in comments below yields the following slightly volume decreasing improvement. Let $A$ be an Apollonian packing of $U'$ which has minimal volume. (Recall $A$ is some infinite maximal disk packing of $U'$). Then we estimate $$\epsilon^*_n=\inf_{A} vol(A),$$ where the infimum is taken over all Apollonian packings $A$ of $U'$.

Obviously $\epsilon_n^*$ is slightly smaller than the volume of $U'$, which is $1/2^{n-1}$ times the volume of the unit disk. Replacing $U'$ by an Apollonian packing $A$ is interesting idea, but does not dramatically reduce the volume.

Question: Is anybody aware of any further progress in Guth's Sponge Problem, or candidate open sets $U\subset \mathbb{R}^n$ with $vol(U)<\epsilon_n^*$ which cannot be expanded embedded into $D$?