[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 $D$?

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