$z=(1.5,-0.9,-0.9,1.1,2.1,1.4)\in {\mathbb{R}}^6$. The interior extrema are ${\left\{ {2,3}\right\}}$ and ${\left\{ {5}\right\}}$. The boundary extrema are ${\left\{ {1}\right\}}$ and ${\left\{ {6}\right\}}$. Thus $\operatorname{cv}(z)\in {{\mathcal{E}}}^3_+$ and $(i_1,i_2,i_3) = (2,5,6)$ or $(i_1,i_2,i_3) = (3,5,6)$. Finally, $h=\operatorname{cv}(z) = (-0.9,2.1,1.4)\in{{\mathcal{E}}}^3_+$.

As is discussed in Section \[sec:background\], if $\operatorname{cv}(z) = \operatorname{cv}(z')$, then ${\mathsf{Dgm}}(z) = {\mathsf{Dgm}}(z')$. Given $h\in{{\mathcal{E}}}_+$, set $$\label{eqCh}
C(h) := {\left\{ {z\in{\mathbb{R}}^N : \operatorname{cv}(z) = h}\right\}}.$$ Then, ${\mathsf{Dgm}}$ is constant on $C(h)$.

Using results of [@curry] in Section \[sec:background\] we prove the following result.

Let ${\mathsf{diag}}\in{\mathsf{Per}}$. Then the preimage of ${\mathsf{diag}}$ is composed of a finite number of mutually disjoint components $${\mathsf{Dgm}}^{-1}({\mathsf{diag}}) = \bigcup_{i=1}^{I({\mathsf{diag}})} C(h^i)$$ where $h^i\in {{\mathcal{E}}}_+$ and all the $h^i$, $i=1,\ldots, I({\mathsf{diag}})$ are related by permutations of the coordinates.

The main result of this paper is as follows.

\[thm:main\] Assume $z\in{\mathbb{R}}^N$ satisfies assumption [**A**]{}. Let $${\mathsf{Dgm}}^{-1}({\mathsf{Dgm}}(z)) = \bigcup_{i=1}^{I({\mathsf{Dgm}}(z))} C(\operatorname{cv}(z)^i).$$ Then for each $i$, $C(\operatorname{cv}(z)^i)$ is an union of a finite *contractible* set of *polytopes*.

Moreover, let ${\mathcal{B}}= \left\{z\in{\mathbb{R}}^N\colon z_1\in[-\bar{a},\bar{a}],\ z_N\in[-\bar{b},\bar{b}]\right\}$ for some $\bar{a},\bar{b} > 0$, and $z\in{\mathcal{B}}$ satisfies assumption [**A**]{}, then $C(\operatorname{cv}(z)^i)\subset{\mathcal{B}}$ is a *compact simplicial set*.

As is described below the proof of Theorem \[thm:main\] occupies the majority of this paper (Sections \[sec:background\] through \[seccontr\]). In Section \[preimageregions\] we prove Theorem \[thm:mainNdelta\]. 