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| Lifting inequalities for polytopes |
| Abstract: The f-vector enumerates the number
of faces of a convex polytope according to dimension. The flag f-vector
is a refinement of the f-vector since it enumerates face incidences of the
polytope. To classify the set of flag f-vectors of polytopes is an open problem
in discrete geometry. This was settled for 3-dimensional polytopes by
Steinitz a century ago. However, already in dimension 4 the problem is open. We will discuss the known linear inequalities for the flag f-vector of polytopes. These inequalities include the non-negativity of the toric g-vector, that the simplex minimizes the cd-index, and the Kalai convolution of inequalities. We will introduce a method of lifting inequalities from lower dimensional polytopes to higher dimensions. As a result we obtain two new inequalities for 6-dimensional polytopes. The talk will be accessible to a general audience. |