Project Details
Projekt
Extremal bodies with respect to lattice functionals
Subject Area
Mathematics
Term
since 2024
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 539867386
We will investigate recently introduced classes of convex bodies in relation to a point lattice, namely lattice reduced and lattice complete bodies. These bodies, defined by two of the applicants, give us an innovative tool to approach the problems listed below, all related to maximizing lattice width. Combining our novel viewpoint of lattice reduced bodies with established techniques promises substantial breakthroughs. 1) Flatness problem: Maximize lattice width of convex bodies without interior integer points. We want to determine explicit maximizers in low dimensions or for special classes of convex bodies by classifying lattice reduced bodies. 2) Flatness problem restricted to lattice polytopes. We want to apply inequalities obtained for general bodies in the previous objective to the enumeration problem of lattice polytopes without interior integer points in small dimension. 3) Maximum lattice width of a lattice polytope with k interior lattice points. We will approach this question computationally for k small as well as asymptotically. 4) "Generalized flatness constant" and "lattice size". A number of different lattice problems can each be expressed by forbidding appropriate structures within a convex body. We aim to set up a unifying framework to synergistically study these problems together.
DFG Programme
Priority Programmes
Subproject of
SPP 2458:
Combinatorial Synergies
Co-Investigator
Professor Dr. Christian Haase
