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Creators/Authors contains: "Lytchak, Alexander"

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  1. ; ; ; (, Journal of the European Mathematical Society)
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  2. ; (, Israel Journal of Mathematics)
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  3. ; (, Journal of Optimization Theory and Applications)
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  4. ; (, Journal für die reine und angewandte Mathematik (Crelles Journal))
    Abstract We give a necessary condition on a geodesic in a Riemannian manifold that can run in some convex hypersurface.As a corollary, we obtain peculiar properties that hold true for every convex set in any generic Riemannian manifold ( M , g ) {(M,g)} .For example, if a convex set in ( M , g ) {(M,g)} is bounded by a smooth hypersurface, then it is strictly convex. 
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  5. ; (, Proceedings of the American Mathematical Society)
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  6. ; ; (, Journal of the European Mathematical Society)
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  7. ; ; (, Journal für die reine und angewandte Mathematik (Crelles Journal))
    Abstract The Dehn function measures the area of minimal discs that fill closed curves in a space; it is an important invariant in analysis, geometry, and geometric group theory. There are several equivalent ways to define the Dehn function, varying according to the type of disc used. In this paper, we introduce a new definition of the Dehn function and use it to prove several theorems. First, we generalize the quasi-isometry invariance of the Dehn function to a broad class of spaces. Second, we prove Hölder extension properties for spaces with quadratic Dehn function and their asymptotic cones. Finally, we show that ultralimits and asymptotic cones of spaces with quadratic Dehn function also have quadratic Dehn function. The proofs of our results rely on recent existence and regularity results for area-minimizing Sobolev mappings in metric spaces. 
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