  We consider the diagonal limit of the conformal bootstrap in arbitrary
dimensions and investigate the question if physical theories are given in terms
of cyclic polytopes. Recently, it has been pointed out that in $d=1$, the
geometric understanding of the bootstrap equations for unitary theories leads
to cyclic polytopes for which the faces can all be written down and, in
principle, the intersection between the unitarity polytope and the crossing
plane can be systematically explored. We find that in higher dimensions, the
natural structure that emerges, due to the inclusion of spin, is the weighted
Minkowski sum of cyclic polytopes. While it can be explicitly shown that for
physical theories, the weighted Minkowski sum of cyclic polytopes is not a
cyclic polytope, it also turns out that in the large conformal dimension limit
it is indeed a cyclic polytope. We write down several analytic formulae in this
limit and show that remarkably, in many cases, this works out to be very good
approximation even for $O(1)$ conformal dimensions. Furthermore, we initiate a
comparison between usual numerics obtained using linear programming and what
arises from positive geometry considerations.
