Extension Complexity of Polytopes with Few Vertices or Facets

We study the extension complexity of polytopes with few vertices or facets. On the one hand, we provide a complete classification of $d$-polytopes with at most $d+4$ vertices according to their extension complexity: Out of the super-exponentially many $d$-polytopes with $d+4$ vertices, all have extension complexity $d+4$ except for some families of size $\theta(d^2)$. On the other hand, we show that generic realizations of simplicial/simple $d$-polytopes with $d+1+\alpha$ vertices/facets have extension complexity at least $2 \sqrt{d(d+\alpha)} -d + 1$, which shows that for all $d>(\frac{\alpha-1}{2})^2$ there are $d$-polytopes with $d+1+\alpha$ vertices or facets and extension complexity $d+1+\alpha$.

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