Geometric Optimization

The basic problem in geometric optimization is to minimize length, area, or some other quantity, among curves or surfaces satisfying a given constraint. A well-known example is that a circle has least perimeter among curves enclosing a given area. Some problems that the REU participants could research are: (1) the octahedron problem which asks what is the least area surface spanning the edges of the octahedron. The conjectured solution is a beautiful, piecewise-planar soap film consisting of twelve triangles and six kites; (2) large minimizing networks in the hyperbolic plane; that is finding the shortest path connecting a set of points in the hyperbolic plane; (3) Melzak's conjecture which asks what polyhedron with unit volume in Euclidean 3-space has the shortest edgelength (for example, a cube with volume 1 has total edge length 12; there is a polyhedron that has a shorter edgelength--can you figure it out?).

Some nice internet sites are:
Soap bubbles and isoperimetric problems (http://math.berkeley.edu/~hutching/pub/bubbles.html)

Exploratorium soap bubble page (http://www.exploratorium.edu/ronh/bubbles/bubbles.html)

Some open problems in soap bubble geometry (http://torus.math.uiuc.edu/jms/Papers/foams/soap-prob.pdf)

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