论文标题
通过图的厚嵌入，一种广义的等级不等式
A Generalized Isoperimetric Inequality via Thick Embeddings of Graphs
论文作者
论文摘要
我们证明了域差异为与$ k $ dialation填充体积的球体的全面等级不等式。假设$ u $是$ \ mathbb {r}^n $ diffemorphic in Euclidean $ n $ -ball中的开放式集合。我们表明，在至少4个维度中，有一张来自标准的欧几里得半径的地图，约$ $ vol（\ partial u）^{1/（n-1）} $至$ u $，在边界上具有1度，$（n-1）$ - 仅根据$ n $而受到一些常数的限制。我们还在开放式集合的维度3中举例说明了一个小$（n-1）$ - 扩张的地图。广义的等法不等式降低为厚度嵌入图的定理，使用kolmogorov-barzdin定理和最大流量最小定理证明了图的厚度。维度3中反例的证明依赖于coarea的不等式和短绕组数量计算。
We prove a generalized isoperimetric inequality for a domain diffeomorphic to a sphere that replaces filling volume with $k$-dilation. Suppose $U$ is an open set in $\mathbb{R}^n$ diffeomorphic to a Euclidean $n$-ball. We show that in dimensions at least 4 there is a map from a standard Euclidean ball of radius about $vol(\partial U)^{1/(n-1)}$ to $U$, with degree 1 on the boundary, and $(n-1)$-dilation bounded by some constant only depending on $n$. We also give an example in dimension 3 of an open set where no such map with small $(n-1)$-dilation can be found. The generalized isoperimetric inequality is reduced to a theorem about thick embeddings of graphs which is proved using the Kolmogorov-Barzdin theorem and the max-flow min-cut theorem. The proof of the counterexample in dimension 3 relies on the coarea inequality and a short winding number computation.
