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Isoperimetric inequality
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== In Hadamard manifolds == [[Hadamard manifold]]s are complete simply connected manifolds with nonpositive curvature. Thus they generalize the Euclidean space <math>\R^n</math>, which is a Hadamard manifold with curvature zero. In 1970's and early 80's, [[Thierry Aubin]], [[Mikhail Leonidovich Gromov|Misha Gromov]], [[Yuri Burago]], and [[Viktor Zalgaller]] conjectured that the Euclidean isoperimetric inequality :<math>\operatorname{per}(S)\geq n \operatorname{vol}(S)^{(n-1)/n}\operatorname{vol}(B_1)^{1/n}</math> holds for bounded sets <math>S</math> in Hadamard manifolds, which has become known as the [[Cartan–Hadamard conjecture]]. In dimension 2 this had already been established in 1926 by [[André Weil]], who was a student of [[Jacques Hadamard|Hadamard]] at the time. In dimensions 3 and 4 the conjecture was proved by [[Bruce Kleiner]] in 1992, and [[Chris Croke]] in 1984 respectively.
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