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KOI Combinatorics Lectures

CB 114
Speaker(s) / Presenter(s):
Lei Xue, University of Michigan

KOI Combinatorics Lectures

Speaker:  Lei Xue, University of Michigan

Title:  A proof of Grünbaum's Lower Bound Conjecture for polytopes, lattices, and strongly regular pseudomanifolds


In 1967, Grünbaum conjectured that any d-dimensional polytope with d+s ≤ 2d vertices has at least φk(d+s, d) = {d+1 choose k+1} + {d choose k+1} - {d+1-k \choose k+1} \] k-faces. In the talk, we will prove this conjecture and discuss equality cases. We will then extend our results to lattices with diamond property (the inequality part) and to strongly regular normal pseudomanifolds (the equality part). We will also talk about recent results on d-dimensional polytopes with 2d+1 or 2d+2 vertices.

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