A cyclic polytope $C(n, d)$ is defined as the convex hull of $n$ distinct points on the moment curve in $\mathbb{R}^d$ (here $n>d$). This is a simplicial polytope so its boundary $\partial C(n, d)$ is a simplicial complex. I would like to know for what values of $n$ the boundary is a flag complex (i.e., a simplicial complex completely determined by its $1$-skeleton).

  • 1
    $\begingroup$ degenerate simplexes only. $\endgroup$ Mar 31 '17 at 20:58

The answer is "never" (except in the obvious case $d=2$, $n\ge 4$).

$C(n,d)$ is neighborly, meaning that every $d/2$ or less vertices define a simplex. In particular, for $d\ge 4$ its graph is complete and the boundary complex cannot be flag. For $d=3$, the boundary complex contains the three edges $1i$, $in$ and $1n$ but not the triangle $1in$, for every $i\in \{3,...,n-2\}$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.