Just a link to Vazirani's guest post on Noam Nisan's blog Algorithmic Game Theory, which seems to relate to my interest in convex sets with beautiful structure.  (Homogeneous self-dual cones, like the positive semidefinite matrices (semidefinite programming) or the positive orthant (linear programming);  just plain homogeneous cones; hyperbolicity cones...; just plain self-dual cones; weakly self-dual cones (ones isomorphic to their dual, but for which no inner product on the space can make the dual (defined according to said inner product) equal to the cone itself), etc...).  And a thought, perhaps misguided:  isn't a crucial ingredient of an efficient algorithm for a convex program usually something like an efficient algorithm for something like: determining membership in the convex set, or calculating a nice barrier function for the set?  And doesn't the ability to do that usually depend on the set (and/or the cone it generates) having a "nice" structure?  Perhaps the detailed properties of this structure bears some relation to the "combinatorial" structure of algorithms like the Ford-Fulkerson algorithm for max-flow?  Or perhaps particular problems have additional structure that isn't being captured in a typical convex formulation.  I'm no expert here (and have indeed forgotten what the Ford-Fulkerson algorithm is, though I think I once read about it in a book by Vazirani).  This post is primarily to remind myself to look at this more closely---and to link to Vazirani's, which seems worth of attention.