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Questions about Hausdorff measures, their variants (such as spherical Hausdorff measures) and generalisations.

20 votes
Accepted

Hausdorff measure and the volume form

I guess you know that it is true in $\mathbb R^k$. Without loss of generality we can assume that $f\geqslant 0$. Fix $\varepsilon>0$ and cover your manifold by $(1\mp\varepsilon)$-Lipschitz charts. Br …
Anton Petrunin's user avatar
1 vote
Accepted

Recovering the length metric from Hausdorff measure

Nan Li proved that it holds for a pair of Alexandrov spaces without boundary; in particular, it solves the problem for Riemannian manifolds. See Lipschitz-Volume rigidity in Alexandrov geometry. It se …
Anton Petrunin's user avatar