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mg.metric geometry – Status of Larry Guth’s Sponge Problem

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

Larry Guth’s *Sponge Problem* asks: Does there live a ceaseless $epsilon=epsilon_n$ such that each launch subset $Usubset mathbb{R}^n$ satisfying $vol(U)< epsilon_n$ admits an increasing embedding $f: Uhookrightarrow D^n$?

Recall $f$ is increasing embedding iff the symmetric matrix ${}^tDfcdot Df$ has all eigenvalues $geq 1$. Equivalently iff $||D_xf(v)||geq ||v||$ for each $xin U$ and tangent vector $vin T_x U$. Equivalently iff $f$ will increase the size of all curves in $U$.

My greatest appraise $epsilon_n^*$ for $epsilon_n$ comes from patent instance of two disks of radius $1/2$ kissing in $D$, which disks can’t breathe expanded embedded into $D$. Thus I appraise $$epsilon^*_2=2 (pi (1/2)^2)=pi^2/4 ,$$ and extra usually $$epsilon_n^*=2 cdot textual content{Vol}(D(0,1/2))=frac{pi^{n/2}}{2^{n-1} cdotGamma(n/2+1)} ,$$ the place $D(0,1/2)$ is the $n$-ball of radius $1/2$.

I’ve been thinking about Sponge drawback for just a few years, and for all that point I’ve but to better on the appraise $epsilon^*_n = epsilon_n$. No signify how “spongy” or “swiss cheesed” the open set $U$ is, only the existence of too many large disjoint disks appears to be obstruction to being expanded-embedded into $D$.

**Question**: Is anyone cognizant of any progress in Guth’s Sponge Problem, or candidate launch units $Usubset mathbb{R}^n$ with $vol(U)<epsilon_n^*$ which can’t breathe expanded embedded into $D$?

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