Compartmentalized structures

• Applicant: Professor Dr. Raman Sanyal
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 539866395

A full-dimensional convex polytope is called unconditional if it is symmetric with respect to reflections in the coordinate hyperplanes. An unconditional polytope can be completely recovered from the restriction to any orthant and these restrictions are characterized by a geometric property called `anti-blocking'. Locally anti-blocking polytopes dispose of the symmetry property by requiring only that the restriction to every orthant is anti-blocking. The resulting subdivision into possibly different anti-blocking polytopes is a basic example of a compartmentalized structure. More generally, a pure fan is a collection of equi-dimensional polyhedral cones that pairwise meet in faces. The fan is complete if the cones cover the ambient space. A polytope is compartmentalized with respect to a complete fan if the restriction to every cone satisfies an anti-blocking-type condition. Many important classes of convex polytopes are compartmentalized, including inscribed or ideal-hyperbolic polytopes, convex bodies invariant under a reflection group as well as orbit polytopes. Moreover, for important classes of fans compartmentalized polytopes are closed with respect to several natural operations such as taking intersections, polars, convex hulls, and Minkowski sums. The goal of the project is to develop the algebraic, combinatorial, and geometric aspects of general compartmentalized structures with a view towards applications. Unconditional and locally anti-blocking polytopes are ideal testing grounds for conjectures including the Mahler conjecture, Godbersen's conjecture, and Kalai's 3^d-conjecture. In the first part of the project, these conjectures will be pursued for general compartmentalized polytopes. In particular closure with respect to polarity allows us to investigate reflexiveness among compartmentalized lattice polytopes and arithmetic generalizations of Mahler's conjecture. Results and techniques developed for compartmentalized polytopes will be adapted to compartmentalized structures for which the underlying fan is not necessarily complete. Minkowski sums are the starting point for a vast theory of geometric inequalities that manifest in many important algebraic structures. In the second part of the project the ramifications of Minkowski sums for non-convex compartmentalized structures will be developed. The resulting algebraic structures together with the accompanying theory of geometric inequalities subsume the so-called normal complexes that give a geometric perspective on the Hodge theory of matroids.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2458: Combinatorial Synergies