The centered polar of C(n,d) is nonempty, bounded, with diameter at most n-d

For all $n>d\ge1$, the centered polar $\mathrm{cyclicPolar}(n,d)$ of the cyclic polytope $C(n,d)$ is nonempty and bounded, and $\mathrm{DiamLE}(\mathrm{cyclicPolar}(n,d),\,n-d)$ holds: its combinatorial diameter is at most $n-d$, for all $n>d$, not only the balanced range $n\le2d$.

This is the Hirsch bound itself, proved for the specific, explicit cyclic-polar family (as opposed to the general Hirsch conjecture, long known false in general by Santos 2012, or the polynomial Hirsch conjecture, still open). The upper bound alone is stated here; the matching exact lower bound in the balanced range is the separate leaf cyclic_polar_diameter_balanced.