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• W2945383960 abstract "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 $W^1$), then 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 $$ W_1(f_+, f_-) cdot mathcal{H}^{d-1}left{x in (0,1)^d: f(x) = 0 right} gtrsim_{d} left( frac{|f|_{L^1}}{|f|_{L^{infty}}} right)^{4 - frac1d} |f|_{L^1} , .$$ 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." @default.
• W2945383960 created "2019-05-29" @default.
• W2945383960 creator A5031661439 @default.
• W2945383960 creator A5054778975 @default.
• W2945383960 date "2019-05-17" @default.
• W2945383960 modified "2023-09-27" @default.
• W2945383960 title "Transport and Interface: an Uncertainty Principle for the Wasserstein distance" @default.
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