Rectifiability of planes and Alberti representations

Guy C. David, Bruce Kleiner

Research output: Contribution to journalArticlepeer-review


We study metric measure spaces that have quantitative topological control, as well as a weak form of differentiable structure. In particular, let X be a pointwise doubling metric measure space. Let U be a Borel subset on which the blowups of X are topological planes. We show that U can admit at most 2 independent Alberti representations. Furthermore, if U admits 2 Alberti representations, then the restriction of the measure to U is 2-rectifiable. This is a partial answer to the case n = 2 of a question of the second author and Schioppa.

Original languageEnglish (US)
Pages (from-to)723-756
Number of pages34
JournalAnnali della Scuola Normale Superiore di Pisa - Classe di Scienze
Issue number2
StatePublished - 2019

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Mathematics (miscellaneous)


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