Lecture – Geometry of Convex Polytopes

When	summer semester 2026 every Wednesday morning April 15th, 2026 – June 17th, 2026
Where\(\;\;\)	MPI Leipzig room to be announced

This course is conceived as a follow up on the block course “Convex Polytopes” by Bernd Sturmfels, though participation in the latter is not a prerequisite. You can find recordings of the block course behind the link.

Description

Much of the appeal of polytope theory comes from the fact that a convex polytope is both a geometric and a combinatorial object. This lecture will focus on the geometric aspects of this interaction, to which there are two approaches:

• the study of the metric properties of individual polytopes, such as edge lengths, dihedral angles, face shapes and volume, the relations between these quantities as well as the constraints imposed on them by the combinatorics.
• the study of the geometric/topological properties of entire realization spaces, that is, the study of the ways in which a fixed combinatorial type can be realized or a given realizations can be deformed.

We shall follow both of these routes in four interconnected sections.

In the first part we take a look at the realization space of a convex polytope. We recall its descriptions as a semialgebraic set, discuss properties and the complexity one faces for high-dimensional polytopes. Eventually we prove the infamous universality theorem.

In the second part we consider polytopes specifically in dimension three. We first reprove Steinitz’ theorem on the combinatorial characterization of 3-polytopes, but in a way that reveals that the realization space (modulo isometries) is homeomorphic to \(\Bbb R^E\), where \(E\) is the number of edges of the polytope. The main tools employed are Tutte embeddings and the Maxwell-Cremona correspondence, of which we give a detailled account. All known constructions of this homeomorphism are highly non-canonical, which raises the question whether canonical parametrizations even exist. Partial solutions are known from Teichmüller theory, rigidity theory, the theory of circle packings, etc., and we will discuss some of them.

In the third part we approach realization spaces from the perspective of rigidity theory: what metric assumptions guarantee that the realization space (modulo isometries) has isolated points? We introduce the core concepts from rigidity theory, including the first-order analysis. We give an overview of the vaste landscape of rigidity results and conjectures, including theorems by Cauchy, Dehn, Gluck, Alexandrov, Whiteley, Connelly, Gortler, Minkowski and Stoker. We discuss the open problem of spherical Tutte embeddings and draw parallels to both canonical parameters and Wachspress Geometry.

In the last part we discuss the question of metric reconstruction of polytopes from partial combinatorial and metric data. We use tools from Wachspress Geometry (Wachspress coordinates, the Izemstiev matrix and the adjoint) to achieve partial reconstruction results.