Alexandrov's embedding theorem and the meaning of realizable

[IBphysicsy]

I am a graduate student working on an algorithm for isometric embeddings of simplicial complexes homeomorphic to $\mathcal{S}^2$with positive Gaussian curvature at each vertex(i.e. deficit angles > 0). Given the constraint on Gaussian curvature, this complex can be regarded as a convex polyhedral metric. Alexandrov's embedding theorem states,

Every convex polyhedral metric can be realized (and in one way only) by a convex polyhedron.

Alexandrov's presentation of his proof is long and cumbersome making it difficult to understand how he draws his conclusion. Because of this, I'm not sure what he means by realizable in this context. Does it mean the polyhedral metrics are identical, or does realizable allow for changes in connections between vertices thus giving a polyhedron that represents the polyhedral metric but is not identical to it? By identical I mean the connections between vertices and edge lengths are the same. If there is anyone who has studied this theorem extensively who can give a precise definition of realizable or give me references concisely summarizing the theorem that would be great. My references are,

author: A.D. Alexandrov title: A.D. Alexandrov Selected Works pt. II Intrinsic Geometry of Convex Surfaces section: Chapter 6 and 7

author: A.V. Pogorelov title: Extrinsic Geometry of Convex Surfaces section: Chapter 1 pg. 20

 

[jvicens]

Dear fellow scientist,

Thank you for reaching out with your question about Alexandrov's embedding theorem. I am also a researcher in the field of geometry and am familiar with this theorem. To answer your question, "realizable" in this context means that the polyhedral metric can be realized or represented by a convex polyhedron. This means that the connections between vertices and the edge lengths are the same as the polyhedral metric being studied.

To clarify, Alexandrov's proof shows that every convex polyhedral metric can be represented by a specific convex polyhedron, with the same connections and edge lengths. This is what is meant by "in one way only." This means that there is only one unique convex polyhedron that can represent a given convex polyhedral metric.

I understand that Alexandrov's proof can be complex and difficult to understand. I would recommend referring to the references you have listed, as well as other resources such as textbooks and research articles, to gain a better understanding of the theorem. Additionally, you can reach out to other experts in the field for further clarification.

I hope this helps clarify the concept of "realizable" in the context of Alexandrov's embedding theorem. Best of luck with your research!