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
κ≤0, then does it have the least boundary when the curvature equals κ
and Ω is round? This conjecture was proven when n = 2 by Weil and
Bol; when n = 3 by Kleiner, and when n = 4 and κ = 0 by Croke. In
joint work with Benoit Kloeckner, we generalize Croke's result to most
of the case κ < 0, and we establish a theorem for κ > 0. 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.