Random Lattice Polytopes

• Applicant: Professor Dr. Matthias Reitzner
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 539849618

The study of lattice polytopes and their properties is a highly active area of research, having its roots in many classical questions. The question, how a generic lattice polytope looks like, i.e. about the average behavior of algebraic and combinatorial properties of lattice polytopes, is important in many recent investigations and are motivated by several applications in different areas in mathematics and computer science. Particular focus in research is on their f-vector. To answer the questions on the generic behavior, random lattice polytopes have been investigated. Classical models are either choosing randomly a lattice polytop out of a set of given lattice polytopes, or intersecting a lattice with a random set and taking the convex hull. The intersection of the integer lattice with a random convex set, and the integer polytope generated by the points of intersection are in the focus of this proposal. We consider the integer lattice and choose a random convex set whose shape is given but its position is chosen at random. Alternatively, one can fix the convex set and choose a randomly shifted and rotated copy of the integer lattice. The convex hull of the lattice points in the intersection yields a random lattice polytope whose combinatorial and metric structure is complicated in general. We are interested in the limiting structure of the random lattice polytope as the size of the convex set tends to infinity. In particular, we would like to analyze the combinatorial and metric behavior of the random lattice polytopes and how these depend on the shape of the underlying convex set. We plan to investigate the asymptotic behavior of the expected f-vector of the random lattice polytope, the expected difference of the intrinsic volumes of the random lattice polytope and the underlying convex set, and the expected Hausdorff distance of the random lattice polytope and the underlying convex set. This leads to the question of determining the random number of lattice points in huge yet very thin 'caps'. Further investigations will deal with higher moments and questions concerning distributional properties of functionals of the random lattice polytopes.

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