Monika Ludwig

Monika Ludwig (Technische Universität Wien, Austria)

Monika Ludwig is Professor of Convex and Discrete Geometry at the Technische Universität Wien (Vienna, Austria). Her field of specialisation is convex geometry and related topics. Ludwig has made significant contributions to the study of convex bodies, geometric valuation theory and discrete geometry.

She is member of the Austrian Academy of Sciences, Fellow of the AMS (inaugural class 2012) and was a plenary lecturer at the European Congress of Mathematics (ECM) in Slovenia in 2021.

Ludwig has published her work in high-impact journals such as Annals of Mathematics, JEMS and Duke Math Journal, and her commitment with training of PhD students is second to none. Her research has pioneered the extension of the study of convex body valuation theory to the context of function spaces.

Research field

Geometric valuation theory ─in the Brunn-Minkowski context and in a functional setting─, which can be encompassed within the so-called metric geometry, an area at the intersection of geometry, analysis and convexity theory.

Research project

Geometric valuation theory is an area of research that interacts with aspects of geometry, analysis and probability, among others, and has been instrumental to allow progress in several important problems.

Historically, its first appearance was in Dehn’s solution (1904) to Hilbert’s third problem, showing that, given two polyhedra with equal volume, one of them cannot always be cut into finitely many pieces that yield the other one after reassembling. Valuation theory has been fundamental in the development of convex geometry, where Hadwiger’s theorem has proved to be a cornerstone with important implications in integral geometry.

Hadwiger’s theorem asserts that every continuous rigid motion invariant valuation on the convex bodies of euclidean space is a linear combination of the intrinsic volumes. For example, in 3D every such valuation must be a linear combination of the volume, the surface area, the mean width and the euler characteristic (the latter playing the role of the constant valuation).

The aim of this project is to explore several questions related to valuation theory, and in particular to Hadwiger’s theorem, in different contexts, with a particular emphasis on dual Brunn-Minkowski theory and functional settings.

Members