Let us consider a finite CW complex $X=\cup X_j$ with a given metric, compatible with the topology (maybe a reasonable one coming from some embedding into some $\mathbb{R}^n$). The characteristic maps are the maps $\phi_j^k:\mathbb{S}^{j-1}\to X_{j-1}$ for $k=1,\ldots, N_j$ which prescribe how to glue the boundaries of the $j$-dimensional cells $e_j^1,\ldots,e_j^{N_j}$ to the $(j-1)$-dimensional skeleton of $X$.

Do there always exist maps $\psi_j^k:\mathbb{S}^{j-1}\to X_{j-1}$ which are homotopic to $\phi_j^k$s and Lipschitz? If the answer is negative, in general, under which hypothesis on $X$ could this be true?

NOTES

- If $X$ is actually a simplicial complex, by simplicial approximation this should be true.
- If $X$ is a CAT(k) space, this is probably true by baricentric subdivision.