Résume | Among n-dimensional regions with fixed volume, which one has the least boundary? This question is known as an isoperimetric problem; its nature depends on what is meant by a ”region”. I will discuss variations of an isoperimetric problem known as the generalized Cartan-Hadamard conjecture: If Ω is a region in a complete, simply connected n-manifold with curvature bounded above by k≤ 0, then does it have the least boundary when the curvature equals k and Ω is round? It was originally inspired by the problem of finding the optimal shape of a planet to maximize gravity at a single point, such as the place where the Little Prince stands on his own small planet. |