The Little Prince and Weil's isoperimetric problem
Résumé
Using linear programming methods, we derive various isoperimetric inequalities in 2 and 4-dimensional Riemannian manifolds whose curvature is bounded from above. First, we consider the problem of shaping a small planet inside a non-positivily curved surface so as to maximize the gravity feeled by a fixed observer (the Little Prince). This provides a pointwise inequality which, integrated on the boundary of a domain, yields Weil's theorem asserting that the planar Euclidean isoperimetric inequality is satisfied inside all simply connected, non-positively curved surfaces. Then, generalizing Croke's proof of the dimension 4 version of this result, we obtain similar statements in manifolds satisfying an arbitrary sectional curvature upper bound. Moreover, the method enables us to state all our results under a relaxed curvature condition.
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