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    A Geometric Lower Bound Theorem

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    We resolve a conjecture of Kalai relating approximation theory of convex bodies by simplicial polytopes to the face numbers and primitive Betti numbers of these polytopes and their toric varieties. The proof uses higher notions of chordality. Further, for C^2-convex bodies, asymptotically tight lower bounds on the g-numbers of the approximating polytopes are given, in terms of their Hausdorff distance from the convex body.Comment: 26 pages, 6 figures, to appear in Geometric and Functional Analysi

    Lower bound theorems for general polytopes

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    For a dd-dimensional polytope with vv vertices, d+1≤v≤2dd+1\le v\le2d, we calculate precisely the minimum possible number of mm-dimensional faces, when m=1m=1 or m≥0.62dm\ge0.62d. This confirms a conjecture of Gr\"unbaum, for these values of mm. For v=2d+1v=2d+1, we solve the same problem when m=1m=1 or d−2d-2; the solution was already known for m=d−1m= d-1. In all these cases, we give a characterisation of the minimising polytopes. We also show that there are many gaps in the possible number of mm-faces: for example, there is no polytope with 80 edges in dimension 10, and a polytope with 407 edges can have dimension at most 23.Comment: 26 pages, 3 figure
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