The isoperimetric problem
What shape has the minimum surface area that can enclose any given, fixed volume? This is the isoperimetric problem – a common question studied in this branch of theoretical mathematics. In the classical (or Euclidean) space, this shape is a geodesic ball (shown on the left). But move this into the Heisenberg group, a typical sub-Riemannian manifold, with its very different properties and metrics, and the geodesic ball is no longer optimal (shown on the right). In the Heisenberg group, the isoperimetric problem remains open.
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