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

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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 $$vol(U) which can’t breathe expanded embedded into $$D$$?

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