A sharp necessary condition for rectifiable curves in metric spaces

• DOI: 10.4171/rmi/1216
• Publication Date: 2020-10-06T21:45:41Z
• Authors: Guy C. David; Raanan Schul
• Keywords: Mathematics - Metric Geometry; FOS: Mathematics; Metric Geometry (math.MG); 0101 mathematics; 01 natural sciences

In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane, using a multiscale sum what is now known as \beta -numbers, numbers measuring flatness given scale and location. This work was generalized to \mathbb R^n by Okikiolu, Hilbert space second author, has many variants variety metric settings. Notably, 2005, Hahlomaa gave sufficient condition for subset be contained curve. We prove sharpest possible converse Hahlomaa’s theorem doubling curves, then deduce some corollaries Banach spaces, well Heisenberg group.