SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 3039-3051, January 2020.
Let $f: (0,1)^d \rightarrow \mathbb{R}$ be a continuous function with zero mean and interpret $f_{+} = \max(f, 0)$ and $f_{-} = -\min(f, 0)$ as the densities of two measures. We prove that if the cost of transport from $f_{+}$ to $f_{-}$ is small, in terms of the Wasserstein distance $W_1 (f_+ , f_-)$, then the Hausdorff measure of the nodal set $\left\{x \in (0,1)^d: f(x) = 0 \right\}$ has to be large (``if it is always easy to buy milk, there must be many supermarkets''). More precisely, we show that the product of the $(d-1)$-dimensional volume of the zero set and the Wasserstein transport cost can be bounded from below in terms of the $L^p$ norms of $f$. We apply this “uncertainty principle" to the metric Sturm--Liouville theory in higher dimensions to show that a linear combination of eigenfunctions of an elliptic operator cannot have an arbitrarily small zero set.

中文翻译：

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传输和接口：Wasserstein距离的不确定性原理

SIAM数学分析杂志，第52卷，第3期，第3039-3051页，2020年1月。
令$ f：（0,1）^ d \ rightarrow \ mathbb {R} $是均值为零的连续函数，并解释$ f _ {+} = \ max（f，0）$和$ f _ {-} =- \ min（f，0）$作为两个度量的密度。我们证明，如果从$ f _ {+} $到$ f _ {-} $的运输成本很小，就Wasserstein距离$ W_1（f_ +，f _-）$而言，则节点集的Hausdorff测度$ \ left \ {x \ in（0,1）^ d：f（x）= 0 \ right \} $必须大一些（``如果总是很容易买牛奶，一定要有很多超市'' ）。更准确地说，我们证明了零集的$（d-1）$维体积与Wasserstein运输成本的乘积可以根据$ f $的$ L ^ p $范数从下面限制。我们采用这种“不确定性原则”