{"paper":{"title":"Transport and Interface: an Uncertainty Principle for the Wasserstein distance","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CA","authors_text":"Amir Sagiv, Stefan Steinerberger","submitted_at":"2019-05-17T19:30:40Z","abstract_excerpt":"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^{\\i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.07450","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}