On mixed Hodge–Riemann relations for translation-invariant valuations and Aleksandrov–Fenchel inequalities

DOI10.1142/S0219199721500498zbMath1496.52011arXiv2011.10248OpenAlexW3154706068WikidataQ114072658 ScholiaQ114072658MaRDI QIDQ5102384

Jan Kotrbatý, Thomas Wannerer

Publication date: 6 September 2022

Published in: Communications in Contemporary Mathematics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/2011.10248

zbMATH Keywords

valuationsconvex bodiesAleksandrov-Fenchel inequalityHodge-Riemann relations

Mathematics Subject Classification ID

Inequalities and extremum problems involving convexity in convex geometry (52A40) Mixed volumes and related topics in convex geometry (52A39) Dissections and valuations (Hilbert's third problem, etc.) (52B45) Convex sets in (n) dimensions (including convex hypersurfaces) (52A20)

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Cites Work

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• Integral geometry and valuations. Notes of the lectures given in the advanced course on integral geometry and valuation theory, Barcelona, Spain, September 6--10, 2010. Edited by Eduardo Gallego and Gil Solanes
• A Fourier-type transform on translation-invariant valuations on convex sets
• Hermitian integral geometry
• Aleksandrov-Fenchel inequalities for unitary valuations of degree 2 and 3
• On Hodge-Riemann relations for translation-invariant valuations
• Additive kinematic formulas for flag area measures
• Convolution of convex valuations
• Two combinatorial applications of the Aleksandrov-Fenchel inequalities
• On simple polytopes
• Hodge theory for combinatorial geometries
• Minkowski valuations and generalized valuations
• Kinematic formulas for tensor valuations
• Combinatorial positivity of translation-invariant valuations and a discrete Hadwiger theorem
• The multiplicative structure on continuous polynomial valuations
• Mixed Hodge-Riemann bilinear relations in the linear situation
• Contact measures in isotropic spaces
• Kotrbatý's theorem on valuations and geometric inequalities for convex bodies
• Curvature measures of pseudo-Riemannian manifolds
• A homogeneous decomposition theorem for valuations on convex functions
• Integral geometry of exceptional spheres
• Contact integral geometry and the Heisenberg algebra
• Integral geometry of complex space forms
• Structure of the unitary valuation algebra
• The mixed Hodge-Riemann bilinear relations for compact Kähler manifolds
• Inequalities between mixed volumes of convex sets
• Introduction to the Theory of Valuations
• Log-concavity properties of Minkowski valuations
• Even valuations on convex bodies
• Local 𝐿^{𝑝}-Brunn–Minkowski inequalities for 𝑝<1
• COMBINATORIAL APPLICATIONS OF THE HODGE–RIEMANN RELATIONS
• Unitarily invariant valuations and Tutte’s sequence
• Mixed volumes and the Bochner method
• Convex Bodies The Brunn-MinkowskiTheory
• Notions of convexity
• Description of translation invariant valuations on convex sets with solution of P. McMullen's conjecture