We introduce the so-called doubling metric on the collection of non-empty bounded open subsets of a metric space. Given an open subset U of a metric space X, the predecessor U_∗ of U is defined by doubling the radii of all open balls contained inside U, and taking their union. The predecessor of U is an open set containing U. The directed doubling distance between U and another subset V is the number of times that the predecessor operation needs to be applied to U to obtain a set that contains V. Finally, the doubling distance between open sets U and V is the maximum of the directed distance between U and V and the directed distance between V and U.

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We prove higher regularity properties of inverse mean curvature flow in Euclidean space: A sharp lower bound for the mean curvature is derived for star-shaped surfaces, independently of the initial mean curvature. It is also shown that solutions to the inverse mean curvature flow are smooth if the mean curvature is bounded from below. As a consequence we show that weak solutions of the inverse mean curvature flow are smooth for large times,beginning from the first time where a surface in the evolution is star-shaped.

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No abstract uploaded!