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    Representations of Polytopes

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    Genre
    Thesis/Dissertation
    Date
    2011
    Author
    Dobbins, Michael Gene
    Advisor
    Rivin, Igor
    Committee member
    Futer, David
    Pollack, Richard
    Szyld, Daniel
    Theran, Louis
    Department
    Mathematics
    Subject
    Mathematics
    Combinatorics
    Discrete Geometry
    Permanent link to this record
    http://hdl.handle.net/20.500.12613/1109
    
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    DOI
    http://dx.doi.org/10.34944/dspace/1091
    Abstract
    Here we investigate a variety of ways to represent polytopes and related objects. We define a class of posets, which includes all abstract polytopes, giving a unique representative among posets having a particular labeled flag graph and characterize the labeled flag graphs of abstract polytopes. We show that determining the realizability of an abstract polytope is equivalent to solving a low rank matrix completion problem. For any given polytope, we provide a new construction for the known result that there is a combinatorial polytope with a specified ridge that is always projectively equivalent to the given polytope, and we show how this makes a naturally arising subclass of intractable problems tractable. We give necessary and sufficient conditions for realizing a polytope's interval poset, which is the polytopal analog of a poset's Hasse diagram. We then provide a counter example to the general realizablity of a polytope's interval poset.
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