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Small Extended Formulations for Cyclic Polytopes
We provide an extended formulation of size O(log n)^{\lfloor d/2 \rfloor} for
the cyclic polytope with dimension d and n vertices (i,i^2,\ldots,i^d), i in
[n]. First, we find an extended formulation of size log(n) for d= 2. Then, we
use this as base case to construct small-rank nonnegative factorizations of the
slack matrices of higher-dimensional cyclic polytopes, by iterated tensor
products. Through Yannakakis's factorization theorem, these factorizations
yield small-size extended formulations for cyclic polytopes of dimension d>2
Axioms for the g-vector of general convex polytopes
McMullen's g-vector is important for simple convex polytopes. This paper
postulates axioms for its extension to general convex polytopes. It also
conjectures that, for each dimension d, a stated finite calculation gives the
formula for the extended g-vector. This calculation is done by computer for d=5
and the results analysed. The conjectures imply new linear inequalities on
convex polytope flag vectors. Underlying the axioms is a hypothesised
higher-order homology extension to middle perversity intersection homology
(order-zero homology), which measures the failure of lower-order homology to
have a ring structure.Comment: LaTeX2e. 10 page
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