We define a centrally symmetric analogue of the cyclic polytope and study its facial structure. We conjecture that our polytopes provide asymptotically the largest number of faces in all dimensions among all centrally symmetric polytopes with
vertices of a given even dimension
when
is fixed and n grows. For a fixed even dimension
and an integer
we prove that the maximum possible number of
-dimensional faces of a centrally symmetric
-dimensional polytope
with
vertices is at least
for some
and at most
as
grows.We show that
and conjecture that the bound is best possible.
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