Transport and Interface: an Uncertainty Principle for the Wasserstein distance

2019/05/17 by Sagiv, Amir, Steinerberger, Stefan
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1905.07450

Abstract

Let f: [0,1]^d → 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 W¹), then the nodal set \x ∈ (0,1)^d: f(x) = 0 \ has to be large (`if it is always easy to buy milk, there must be many supermarkets'). More precisely, we show that W₁(f₊, f_-) ⋅ H^d-1\x ∈ (0,1)^d: f(x) = 0 \ \gtrsim_d ( \frac‖f‖_L¹‖f‖_L^∞ )^{4 - \frac1d} ‖f‖_L¹ . 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.

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