Curvature Measures in Convex and Integral Geometry
Final Report Abstract
This project contributed to recent advances in an area of pure mathematics known as valuation theory, which studies how certain mathematical functions—called valuations—can measure the size or structure of geometric objects like polytopes or compact submanifolds. Over the past two decades, researchers have discovered that they possess a rich algebraic structure and deep connections to fields like integral geometry and combinatorics. A central goal of the project was to investigate so-called curvature measures, which are local versions of valuations and play a key role in applications to integral geometry. The project laid the mathematical foundations for a theory of continuous translation-invariant curvature measures on convex bodies. Important foundational results were achieved, including a simplification of the axiomatic framework and a characterization of curvature measures in specific dimensions and degrees. The research also concerned the extension of these concepts to more general spaces known as manifolds—geometric objects that locally resemble Euclidean space. In collaboration with other researchers, new valuations and curvature measures were discovered for Kähler manifolds, which are central objects in modern geometry and theoretical physics. Unexpectedly, the project also contributed to the discovery of a new algebraic structure on the space of valuations, colloquially referred to as a “Kähler package”—a set of deep geometric and algebraic properties originally observed in complex geometry. This structure includes the celebrated Hodge–Riemann relations and the hard Lefschetz theorem, both of which are key features in modern algebraic geometry.
Publications
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On mixed Hodge–Riemann relations for translation-invariant valuations and Aleksandrov–Fenchel inequalities. Communications in Contemporary Mathematics, 24(07).
Kotrbatý, Jan & Wannerer, Thomas
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Curvature measures of pseudo-Riemannian manifolds. Journal für die reine und angewandte Mathematik (Crelles Journal), 2022(788), 77-127.
Bernig, Andreas; Faifman, Dmitry & Solanes, Gil
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From harmonic analysis of translation-invariant valuations to geometric inequalities for convex bodies. Geometric and Functional Analysis.
Kotrbatý, Jan & Wannerer, Thomas
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Hard Lefschetz theorem and Hodge-Riemann relations for convex valuations
Thomas Wannerer, A. Bernig & J. Kotrbatý
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Minkowski Valuations in Affine Convex Geometry. International Mathematics Research Notices, 2025(1).
Henkel, Jakob & Wannerer, Thomas
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Complex and quaternionic analogues of Busemann’s random simplex and intersection inequalities. Mathematische Annalen, 393(2), 1797-1825.
Saroglou, Christos & Wannerer, Thomas
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Dually Lorentzian Polynomials. Monatshefte für Mathematik, 208(3), 495-524.
Ross, Julius; Süss, Hendrik & Wannerer, Thomas
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The Fourier transform on valuations is the Fourier transform. Journal of Functional Analysis, 288(3), 110741.
Faifman, Dmitry & Wannerer, Thomas
