Isoperimetrische Ungleichungen und Minkowski Bewertungen in einem komplexen Vektorraum
Zusammenfassung der Projektergebnisse
The aim of the project was to study geometric inequalities, valuations, and integral geometry in the framework of a complex vector space. More precisely, the principal lines of study where results were obtained are: (1) Isoperimetric type inequalities for hermitian intrinsic volumes. (2) Geometric properties of the complex projection and difference bodies. (3) Characterization results for Minkowski valuations by using as a condition an affine geometric inequality. (4) Integral geometry of flag area measures. (5) Affine tensor valuations in a complex vector space. (1) The first isoperimetric type inequality involving hermitian intrinsic volumes was obtained. A sharp isoperimetric inequality between hermitian intrinsic volumes of degree of homogeneity 2 and 3 was given and shown that this inequality is not satisfied for all such hermitian intrinsic volumes. For instance, the hermitian intrinsic volumes satisfying it are monotone valuations. The obtained isoperimetric inequality is a direct consequence of a sharp Aleksandrov-Fenchel inequality for the involved hermitian intrinsic volumes. A new method (by using hyperbolic polynomials) had to be used to obtain some pointwise inequalities. (2) We obtained geometrical properties and inequalities satisfied by the complex difference body. The basic difference compared to the classical difference body is that the dimension of the complex difference body depends on the position of the body with respect to the complex structure of the vector space. Spherical harmonics were used to obtain geometric properties. (3) The aim was to obtain classification results for Minkowski valuations by using as a main hypothesis the fulfillment of a volume constraint, that is, by imposing that there is an upper and a lower bound for the volume of the image of K in terms of the volume of the convex body K itself. In particular, we proved that in the case of continuous, translation invariant Minkowski valuations satisfying the above volume constraint, only two types of operators appear: a family of operators having only cylinders over (n − 1)-dimensional convex bodies as images, and a second family consisting essentially of 1-homogeneous operators. The case of 1-homogeneous operators was studied and the monotone (resp. rotation invariant) ones classified. For that, a new representation result for such operators being even and symmetric was obtained after establishing the existence of bi-Lipschitz bijections on the Grassmannian Gr(n, k) for every 1 ≤ k ≤ n − 1, associated to every operator in the considered class. Some previous results involving the condition that the operator is GL(n)-covariant, but not necessarily Minkowski valuation were obtained. In particular, a new characterization of the difference body operator was given. (4) A Hadwiger type theorem and additive kinematic formulas were obtained for flag area measures, i.e., smooth translation invariant valuations taking values in the space of signed measures on the flag space consisting of all (p+1)-planes containing a unit vector. The existence of additive kinematic formulas for general flag area measures was also proved and an algebraic framework to compute explicitly these formulas for a given flag manifold was given. (5) The space of continuous, SL(m, C)-equivariant, m ≥ 2, and translation covariant valuations taking values in the space of real symmetric tensors on Cm of rank r ≥ 0 was described. The classification involves only the moment tensor valuation for r ≥ 1 and is analogous to the classification of the corresponding tensor valuations that are SL(2m, R)-equivariant. However, the method of proof cannot be adapted, and we used results from the theory of continuous and translation invariant valuations together with results from representation theory.
Projektbezogene Publikationen (Auswahl)
- Aleksandrov-Fenchel inequalities for unitary valuations of degree 2 and 3. Calculus of Variations and Partial Differential Equations, 54, no. 2, 1767–1791, 2015
Judit Abardia and Thomas Wannerer.
(Siehe online unter https://doi.org/10.1007/s00526-015-0843-0) - How do difference bodies in complex vector spaces look like? A geometrical approach. Communications in Contemporary Mathematics, 17, no. 4, 1450023, 32 pp., 2015
Judit Abardia and Eugenia Saorín Gómez
(Siehe online unter https://doi.org/10.1142/S0219199714500230) - The role of the Rogers-Shephard inequality in the characterization of the difference body. Forum Mathematicum, 29, no. 6, 1227–1243, 2017
Judit Abardia-Evéquoz and Eugenia Saorín Gómez
(Siehe online unter https://doi.org/10.1515/forum-2016-0101) - Minkowski additive operators under volume constraints. The Journal of Geometric Analysis, 28, no. 3, 2422–2455, 2018
Judit Abardia-Evéquoz, Andrea Colesanti, and Eugenia Saorín Gómez
(Siehe online unter https://doi.org/10.1007/s12220-017-9909-x) - Minkowski valuations under volume constraints. Advances in Mathematics, 333, 118–158, 2018
Judit Abardia-Evéquoz, Andrea Colesanti, and Eugenia Saorín Gómez
(Siehe online unter https://doi.org/10.1016/j.aim.2018.05.033) - Flag area measures. Mathematika, 65, 958–989, 2019
Judit Abardia-Evéquoz, Andreas Bernig, and Susanna Dann
(Siehe online unter https://doi.org/10.1112/S0025579319000226) - SL(m, C)-equivariant and translation covariant continuous tensor valuations. Journal of Functional Analysis, 276, no. 11, 3325–3362, 2019
Judit Abardia-Evéquoz, Károly Jr. Böröczky, Mátyás Domokos, and Dávid Kertész
(Siehe online unter https://doi.org/10.1016/j.jfa.2019.02.015) - Additive kinematic formulas for flag area measures
Judit Abardia-Evéquoz and Andreas Bernig
