Sometimes, it is possible to represent a complicated polytope as a projection
of a much simpler polytope. To quantify this phenomenon, the extension
complexity of a polytope P is defined to be the minimum number of facets in a
(possibly higher-dimensional) polytope from which P can be obtained as a
(linear) projection. This notion has been studied for several decades,
motivated by its relevance for combinatorial optimisation problems. It is an
important question to understand the extent to which the extension complexity
of a polytope is controlled by its dimension, and in this paper we prove three
different results along these lines. First, we prove that for a fixed dimension
d, the extension complexity of a random d-dimensional polytope (obtained as
the convex hull of random points in a ball or on a sphere) is typically on the
order of the square root of its number of vertices. Second, we prove that any
cyclic n-vertex polygon (whose vertices lie on a circle) has extension
complexity at most 24nβ. This bound is tight up to the constant factor
24. Finally, we show that there exists an no(1)-dimensional polytope
with at most n facets and extension complexity n1βo(1).Comment: We fixed an issue with Lemma 6.9 (the exponential Efron-Stein
inequality was previously used incorrectly