My research interests are in convex, discrete and combinatorial geometry. A major focus lies on combinatorial problems revolving around lattice points in polyhedra, which are inspired by phenomena in the Euclidean theory of convex bodies. The most basic question of this kind is: Given the volume of a polytope, what can be said about the number of integer points it contains? How do other geometric quantities (such as width or diameter) of the polytope affect the discrepancy of volume and lattice point count? Beyond volume, I studied further concepts from convex geometry that have interesting and non-trivial discretizations, such as hyperplane sections and projections, or equivariant valuations.

Currently, I'm very interested in structural properties of convex bodies that are "extremal" with respect to certain lattice functionals, such as width or covering radius. This line of thought is also inspired by classical geometric ideas such as "bodies of constant width." The goal is to use such structural results to prove tight inequalities on the functionals involved, for instance in the context of the flatness problem in integer linear programming.

Apart from lattice problems, I am also working on other topics in convex geometry such as volume distribution in polytopes or convexity in spaces of constant curvature.

Preprints

• Pal's isominwidth problem in the hyperbolic space , with Károly J. Böröczky and Ádám Sagmeister. [arXiv]
• The isominwidth problem on the 2-sphere, with Ádám Sagmeister. [arXiv]
• Exponential valuations on lattice polygons, with Károly J. Böröczky, Mátyás Domokos, Christoph Haberl, Gergerly Harcos and Jin Li. [arXiv]
• Unimodular valuations beyond Ehrhart, with Monika Ludwig and Martin Rubey. [arXiv]

Publications

• The affine subspace concentration inequality for centered convex bodies, with Katharina Eller.
  accepted in Acta Math. Hung. [journal] [arXiv]
• Lattice reduced and complete convex bodies, with Giulia Codenotti.
  Journal of the LMS 110(4):e12982, 2024. [journal] [arXiv]
• Polynomial bounds in Koldobsky's discrete slicing problem, with Martin Henk.
  Proceedings of the AMS 152:3063-3074, 2024. [journal] [arXiv]
• Affine subspace concentration conditions for centered polytopes, with Martin Henk and Christian Kipp.
  Mathematika 69(2):458-472, 2023. [journal] [arXiv]
• Interpolating between volume and lattice point enumerator with successive minima, with Eduardo Lucas.
  Monatshefte für Mathematik 198:717-740, 2022. [journal] [arXiv]
• Bounds for the lattice point enumerator via sections and projections, with Martin Henk.
  Discrete and Computational Geometry 67:895-918, 2022. [journal] [arXiv]

Thesis

Aspects of volume of convex bodies - discretization, subspace concentration and polarity, doctoral thesis, TU Berlin. [link]