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CURVATURE INTEGRAL AND LIPSCHITZ PARAMETRIZATION IN 1-REGULAR METRIC SPACES

We show that for a bounded 1-regular metric measure space (E,μ) the finiteness of the Menger curvature integral ∫ E ∫ E ∫ E c(z 1 ,z 2 ,z 3 ) 2 dμz 1 dμz 2 dμz 3 guarantees that E is a Lipschitz image of a subset of a bounded subinterval of R.