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Toric varieties and convex polyhedra are intimately connected. Some of this can be found in standard text books (the connection between divisors and mixed volumes seems to be a popular example).

One of the most important objects that are associated to an algebraic variety is its derived category. So I'm wondering: are there any constructions or properties in convex geometry that are reflected in the derived category of associated toric varieties?

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Derived category of a toric variety has a full exceptional collection which was constructed by Kawamata using toric Minimal Model Program. As far as I know no good direct relation between the collection and the fan (polyhedron) of the variety is known.

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