On the Dimensions of the Realization Spaces of Polytopes

TL;DR

This paper investigates the structure and dimension of realization spaces of convex polytopes, confirming Robertson's dimension formula for many classes but also identifying exceptions and complexities revealed by Mnev's universality theorem.

Contribution

It develops Jacobian criteria for analyzing realization spaces, confirming the dimension formula for many classes and identifying specific polytopes where it fails.

Findings

Realization spaces are manifolds with dimension NG(P) for many classes.

Counterexamples like the bipyramid over a triangular prism show failure of the dimension count.

The 24-cell's realization space has complex manifold properties, not smoothly embedded everywhere.

Abstract

Robertson (1988) suggested a model for the realization space of a convex d-dimensional polytope and an approach via the implicit function theorem, which -- in the case of a full rank Jacobian -- proves that the realization space is a manifold of dimension NG(P):=d(f_0+f_{d-1})-f_{0,d-1}, which is the natural guess for the dimension given by the number of variables minus the number of quadratic equations that are used in the definition of the realization space. While this indeed holds for many natural classes of polytopes (including simple and simplicial polytopes, as well as all polytopes of dimension at most 3),and Robertson claimed this to be true for all polytopes, Mnev's (1986/1988) Universality Theorem implies that it is not true in general: Indeed, (1) the centered realization space is not a smoothly embedded manifold in general, and (2) it does not have the dimension NG(P) in general. In this paper we develop Jacobian criteria for the analysis of realization spaces. From these we get easily that for various large and natural classes of polytopes the realization spaces are indeed manifolds, whose dimensions are given by NG(P). However, we also identify the smallest polytopes where the dimension count (2) and thus Robertson's claim fails, among them the bipyramid over a triangular prism. For the property (1), we analyze the classical 24-cell: We show that the realization space has at least the dimension 48, and it has points where it is a manifold of this dimension, but it is not smoothly embedded as a manifold everywhere.