# Quasiconformal map from a subset of $\mathbb{C}$ to a polytope

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?

Thank you.

• Every piecewise linear homeomorphism (from a finite polygon) is quasiconformal. Apr 10 at 22:41
• 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? Apr 10 at 23:39
• Either one will give you a quasiconformal map. As for conformal maps: If your homeomorphism is piecewise-conformal then it is conformal. Apr 10 at 23:50
• Do you have any examples of piecewise-conformal homeomorphisms between polygons (possibly a reference in literature)? Thank you again for all of your insight. Apr 11 at 0:24
• I am not good with explicit examples. Apr 11 at 2:18

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.