On the realization spaces of convex polyhedra

Abstract

In this monograph we investigate the structure of realization spaces of convex polytopes in dimension 3. Given a Euclidean convex polyhedron $\mathbf{P}$ with $e$ edges, we use the celebrated Steinitz's theorem to deduce that the space $\mathcal{R}(P)$ of all realizations of $\mathbf{P}$ is a smooth manifold of dimension $e+6$. By identifying Euclidean affine space $\mathbb{R}^3$ with an affine subspace of the projective sphere $\mathbb{S}^3$, we consider the action of Aut$(\mathbb{S}^3) = SL_{\pm}(4,\mathbb{R})$ on $\mathcal{R}(P)$ to construct the \textit{projective prerealization space} $\mathcal{R}_{\mathbb{P}}(P)$. By identifying the projective equivalence classes of this action, we find that the \textit{projective realization space} $\mathcal{RS}_{\mathbb{P}}(P):= \mathcal{R}_{\mathbb{P}}(P) / \mathbf{SL}(4,\mathbb{R})$ is a smooth manifold of dimension $e-9$.