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Question. Does a quasiconformal map exist between a subset of $\mathbb{C}$ (such as a unit disc or rectangle) and a polytope?

Here, I take a polytope to be a two-dimensional surface that could be embedded in $\mathbb{R}^3$ or some other three-dimensional space, possibly $\mathbb{H}^3$.

In particular, this question interested me as it would offer a method to parameterize a permutohedron with the complex plane. The premise of this question was inspired by Dantzig's method, a numerical optimization technique in linear programming that can be realized as a path along the edge of a simplex.

By segmenting $\mathbb{R}^3$ into a series of planes and applying a Schwarz-Christoffel transform on each, I was able to form an injective map from the unit disc to polytope in three dimensions; however, I do not believe it is quasiconformal. As a secondary question, does anyone know if any such mapping could preserve the holomorphicity of a function on $\mathbb{C}$?

Edit (Clarification): Could it be possible to make a quasiconformal or conformal map from a disk in $\mathbb{C}$ to the following polytope?

A Permutohedron as a Polytope in Three Dimensions

Thank you.

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    $\begingroup$ Every piecewise linear homeomorphism (from a finite polygon) is quasiconformal. $\endgroup$ Apr 10 at 22:41
  • $\begingroup$ Thank you for your prompt response! By piecewise linear homeomorphism from a finite polygon, do you mean a map from a polygon embedded in $\mathbb{C}$ to a polytope embedded in $\mathbb{R}^3$? Or, do you mean a map between polygons in $\mathbb{C}$ or $\mathbb{R}^3$? If I were to ask a follow-up question about purely conformal maps, would it be best practice to make a separate question? $\endgroup$
    – JPwin
    Apr 10 at 23:39
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    $\begingroup$ Either one will give you a quasiconformal map. As for conformal maps: If your homeomorphism is piecewise-conformal then it is conformal. $\endgroup$ Apr 10 at 23:50
  • $\begingroup$ Do you have any examples of piecewise-conformal homeomorphisms between polygons (possibly a reference in literature)? Thank you again for all of your insight. $\endgroup$
    – JPwin
    Apr 11 at 0:24
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    $\begingroup$ I am not good with explicit examples. $\endgroup$ Apr 11 at 2:18
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Yes, there exists a quasiconformal map (even a homeomorphism) from the Riemann sphere to a polytope. Embedding to $R^3$ is irrelevant here, all we need is the intrinsic metric, which is a flat metric with conic singularities. Consider a conic singularity with angle $2\pi\alpha$. Map a neighborhood of $0$ onto a neighborhood of this point by the map which has representation in local coordinates: $f(z)=|z|^{1-\alpha}z^\alpha$. This map is quasiconformal. Define such maps in neighborhoods of all singularities. Then extend it to the whole sphere smoothly.

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