Frank Morgan

Abstract: The version of the shortest “Steiner” network problem in which you minimize length plus number of Steiner points has an interesting analog in manifolds with density.On a Riemannian manifold M, a density f is fundamentally different from a conformal change of metric because for every m, m-dimensional area is weighted by simply f in contrast to different powers of . This difference shows up in any problem that involves objects of different dimensions, such as the isoperimetric problem, which involves both volume and perimeter. Similarly, it shows up in the network problem of minimizing (weighted) length plus the sum of the weights of additional Steiner points. The equilibrium condition at a Steiner point with edges going in unit directions is that

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This problem arose at a lunch at the 2009 annual joint mathematics meeetings in DC with Konrad Swanepoel, Max Engelstein, Scott Greenleaf, Neil Hoffman, and Bret Thacher.