Bounded regularity subdivision conjecture for simplicial complexes

Let $K$ be a $p$-dimensional simplicial complex in ${\mathbb R}^q$, and let $\epsilon>0$. Write $M_\epsilon$ for a subdivision of $K$, and $M_\epsilon'$ for a subcomplex of $M_\epsilon$; let $\upvartheta_{M_\epsilon}$ and $\upvartheta_{M_\epsilon'}$ denote their simplicial regularity constants. The $(p-1)$-skeleton of $K$ is the union of its faces of dimension at most $p-1$.

Bounded regularity subdivision conjecture. There exist $M_\epsilon$ and $M_\epsilon'$ such that

$$M_\epsilon \backslash M_\epsilon' \subseteq \{\mathbf{x}\in{\mathbb R}^q\mid \|\mathbf{x}-\mathbf{y}\|<\epsilon\text{ for some }\mathbf{y}\text{ in the }(p-1)\text{-skeleton of }K\},$$ $$\upvartheta_{M_\epsilon}\leq\alpha_K,$$

and

$$\upvartheta_{M_\epsilon'}\leq\beta,$$

where $\alpha_K$ is independent of $\epsilon$ and $\beta$ is independent of $K$ and $\epsilon$.

This conjecture formalizes the requirement that subdivision can localize irregular simplices near the original complex's $(p-1)$-skeleton. Proving it in greater generality would extend the paper's main results; the supplied text gives no evidence that it has been resolved.