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In this article on extension complexity of regular polygons https://arxiv.org/pdf/1505.08031.pdf it is mentioned that extension complexity of $n$ regular polygons should be $\theta(\log n)$. This is an exponential gap.

  1. Can the gap be more than exponential for some non-trivial classes of polytopes such as $n$ inequalities being replaced by projection of $o(\log n)$ inequalities? What are the best known examples?

  2. If the lower bound is $\alpha\log_2 n-\beta$ what are the sharpest tradeoffs and bounds on $\alpha$ and $\beta$?

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I think that the gap is at most exponential, and this follows from a simple duality argument. (My original answer used the upper bound theorem: thanks to Emil Jeřábek for pointing out that this is completely unnecessary.)

Let $K$ be a polytope with $m$ facets in $d$-dimensional space ($d \le m-1$), whose orthogonal projection on some subspace $W$ of dimension $k$ is the polytope $L$ with $n$ facets. Without loss of generality, we can assume that $K$ and $L$ contain the origin $0$. Let $K^\circ = \{y: \langle y, x \rangle \le 1 \ \ \forall x \in K\}$ be the polar (dual) polytope to $K$. The vertices of $K^\circ$ are exactly the normal vectors to the facets of $K$, so $K^\circ$ has $m$ vertices. Then $$L = \{x: \langle y, x \rangle \le 1 \ \ \forall y \in K^\circ \cap W\},$$ and the vertices of $K^\circ \cap W$ are exactly the normals to the $n$ facets of $L$. (In other words, $L^\circ = K^\circ \cap W$.) Every $(d-k)$-dimensional face of $K^\circ$, when intersected with $W$, gives at most one vertex of $K^\circ \cap W$, so, $n \le f_{d-k}(K^\circ)$, where the right hand side is the number of $(d-k)$-dimensional faces of $K^\circ$. Since any face of $K^\circ$ is the convex hull of the set of vertices it contains, so it is uniquely determined by them, we have $f_{d-k}(K^\circ) \le 2^m$. So, an upper bound is $n \le 2^{m}$, i.e. the extension complexity of a polytope with $n$ facets is at least $\lceil\log_2(n)\rceil$.

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  • $\begingroup$ Excellent if it is $\geq \alpha\log_2n$ can the $\alpha$ be $1-O(1)$? $\endgroup$ – VS. Mar 29 at 21:59
  • $\begingroup$ I think that sounds plausible, and, if true, it must also follow from the upper bound theorem, which bounds the $h$-vector of the polytope. $\endgroup$ – Sasho Nikolov Mar 29 at 22:13
  • $\begingroup$ I feel it is unlikely that either $\alpha\leq1$ or $\alpha=1$ and $\beta>0$ is possible for any finite facet polytope since it seems to me that we make $n$ 'decisions' from $\log_2n$ 'measurements'. I might be wrong. $\endgroup$ – VS. Mar 30 at 0:09
  • $\begingroup$ I think $\lceil-1+\log_2n\rceil$ might be doable leaving no information theoretic gap. However I do not know lower value is allowable although I would be interested in knowing. $\endgroup$ – VS. Mar 30 at 6:12
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    $\begingroup$ I may be missing something, but each face of $K^\circ$ is uniquely determined by the set of its vertices, hence the total number is at most $2^m$ (without using any sophisticated theorem). $\endgroup$ – Emil Jeřábek Mar 30 at 10:43

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