I would like to know if my sequence of continuous paths of tensor product operators has a convergent subsequence, so that I get a continuous path of product operators as a limit.
THE CONTEXT: I am working in a tensor product Hilbert space, say $H=\otimes_{\alpha=1}^PH_\alpha$, with the Hilbert-Schmidt norm to measure distances. We have a set of positive operator-valued measures (POVM), $M_r=\{E_l^{(r)}\}_r$, which by assumption satisfy $\lim_{r\to\infty}M_r=M$, with $M$ itself a POVM, $M=\{E_j\}$. Each of the $E_l^{(r)}$ and $E_j$ is a positive semidefinite tensor product operator, and the individual sets sum to the identity, $\sum_jE_j=I_H=\sum_lE_l^{(r)}$ for all $r$. For each $r$, the set $\{E_l^{(r)}\}_l$ generates a zonotope, $Z_r$, which is closed, bounded and convex, and similarly zonotope $Z$ is generated by the set $\{E_j\}$. For example, $Z$ is the Minkowski sum of line segments $[0,E_1],[0,E_2],\cdots,[0,E_N]$, with $N$ the number of $E_j$, no two of which are proportional to each other.
The paths: The paths in our sequence of paths are as follows: For each $r,l$, there is a path of positive semidefinite tensor product operators on $H$ lying entirely in $Z_r$ and stretching from $I_H$ to $E_l^{(r)}$. One may write this path as $\Pi_r(s)=\sum_lc_l^{(r)}(s)E_l^{(r)}$, where $0\le c_l^{(r)}(s)\le1$, $0\le s\le1$, $\Pi_r(0)=I_H$ and $\Pi_r(1)=E_l^{(r)}$. These paths are piecewise linear and continuous, but their derivatives with respect to $s$ will have $r$ discontinuities over the range $0\le s\le1$.
The question: We have that $\Pi_r(s)\in Z_r$. Since $\lim_{r\to\infty}Z_r=Z$ and $\lim_{r\to\infty}E_l^{(r)}=E_j$, we would like to know if there exists a continuous path, $\Pi(s)\in Z$, of positive semidefinite product operators stretching from $I_H$ to $E_j$.
What I've tried:
Consider the line, $L=(1-x)I_H+xE_j$ from $I_H$ at $x=0$ to $E_j$ at $x=1$. For each $x$ denote the hyper-plane perpendicular to $L$ as $P(x)$, each of which contains an infinity of points $\Pi_r(x)$, and each of those points is a positive semidefinite tensor product operator lying within a closed, bounded set: $0\le\Pi_r(x)\le I_H$. Then by the Bolzano-Weierstrass theorem, for each $x$ there exists a subsequence of points $\{\Pi_r(x)\}_r$ having a limit that lies in $P(x)$ and that is a product operator. However, I do not see how to prove that these limit points constitute a continuous path. Following one way of proving Bolzano-Weierstrass by identifying "peaks" in the sequence to identify a convergent subsequence, I have tried coming up with an argument that shows that the set of peaks evolves continuously with $x$. However, this seems to fail due to the possibility that in the limit, each value of $x$ could have a different set of peaks, so the evolution of these peak-sets is not continuous.
I have also looked at the Arzelà–Ascoli theorem, but there appear to be two problems with using this result in my case. First, my paths are piecewise linear and continuous, but their derivatives are not continuous. In the limit, they will have an unbounded number of discontinuities in their slopes. The second issue is that I do not see any way to bound these derivatives, even if one ignores these slope discontinuities and only looks at the intervening intervals (which seems questionable, though I have seen it suggested that one can: Real analysis, Proof on Uniform Convergence of a subsequence question).
Thanks in advance for any help.