We will consider optimization problems where the unknown is a one-dimensional object (such as a network) constrained by length and optimizing some geometric functional. We will introduce the tools used in the study of the Steiner problem (minimal length network spanning some set of points), the irrigation problem (minimal network which minimizes its distance from a given set of points) and minimal cluster (the minimal interface required to enclose and separate regions of the plane with given area).
The Steiner Problem. Melzak’s construction. Hausdorff one-dimensional measure, Hausdorff distance, Golab’s theorem.
Irrigation problems, duality formulation. Generalized Steiner Problem.
Minimal clusters: existence & Regularity.
The double bubble, weak formulation. Moebius transformation, monotonicity. The honeycomb problem.