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title: "Characterization of Rectifiable Measures that are Carried by Lipschitz Graphs2"
collection: publications
permalink: /publication/lipschitz
excerpt: 'The Analysts' Traveling Salesman Problem asks for necessary and sufficient conditions under which a set is contained inside of a Lipschtiz image. One direction for further study is to find a characterization of measures carried by Lipschitz graphs. In previous work, balls centered at each point in the support are used to give a characterization of doubling measures that are carried by Lipschitz graphs. To further extend that work, we develop and prove sufficient and necessary conditions for doubling measures carried by Lipschitz graphs in terms of dyadic cubes. Along the way, we prove a doubling measure property and a geometric lemma for measures that hold under the dyadic cube regime. These new results provide a characterization of measures carried by Lipschitz graphs that is more discrete in nature.'
date: 2022-04-08
venue: '2022 Virtual Joint Mathematics Meetings (JMM 2022)'
paperurl: 'https://zcczhang.github.io/files/Geometric_Measure_Theory_Research.pdf'
citation: 'Zhang, Zichen, and Yutong Wu. "Characterization of Rectifiable Measures Carried by Lipschitz Graphs." 2022 Virtual Joint Mathematics Meetings (JMM 2022). AMS.'
---

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The Analysts' Traveling Salesman Problem asks for necessary and sufficient conditions under which a set is contained inside of a Lipschtiz image. One direction for further study is to find a characterization of measures carried by Lipschitz graphs. In previous work, balls centered at each point in the support are used to give a characterization of doubling measures that are carried by Lipschitz graphs. To further extend that work, we develop and prove sufficient and necessary conditions for doubling measures carried by Lipschitz graphs in terms of dyadic cubes. Along the way, we prove a doubling measure property and a geometric lemma for measures that hold under the dyadic cube regime. These new results provide a characterization of measures carried by Lipschitz graphs that is more discrete in nature.

Recommended citation: Zhang, Zichen, and Yutong Wu. "Characterization of Rectifiable Measures Carried by Lipschitz Graphs." 2022 Virtual Joint Mathematics Meetings (JMM 2022). AMS.