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We prove that an isoparametric hypersurface with four principal curvatures and multiplicity pair (7,8) is either the one constructed by Ozeki and Takeuchi, or one of the two constructed by Ferus, Karcher, and Mu¨nzner. This completes the classification of isoparametric hypersurfaces in spheres that ´E. Cartan initiated in the late 1930s.

We show that the inner distance inside a bounded planar domain is at most the one-dimensional Hausdorff measure of the boundary of the domain. We prove this sharp result by establishing an improved Painlevé length estimate for connected sets and by using the metric removability of totally disconnected sets, proven by Kalmykov, Kovalev, and Rajala. We also give a totally disconnected example showing that for general sets the Painlevé length bound ϰ(E)≤πH^1(E)is sharp.

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