Aleksandrov-Fenchel inequalities for unitary valuations of degree $$2$$ 2 and $$3$$ 3

Abardia, Judit; Wannerer, Thomas

doi:10.1007/s00526-015-0843-0

Cited by 13 publications

(17 citation statements)

References 54 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To explain the idea of the proof, we first consider the case k = 1. For h > 0, let u h = 1] . Since Z is a translation invariant valuation and by (19), we obtain…”

Section: Valuations On Cone and Indicator Functionsmentioning

confidence: 99%

Valuations on Convex Functions

Colesanti

Ludwig

Mussnig

2017

International Mathematics Research Notices

View full text Add to dashboard Cite

show abstract

“…To explain the idea of the proof, we first consider the case k = 1. For h > 0, let u h = 1] . Since Z is a translation invariant valuation and by (19), we obtain…”

Section: Valuations On Cone and Indicator Functionsmentioning

confidence: 99%

Valuations on Convex Functions

Colesanti

Ludwig

Mussnig

2017

International Mathematics Research Notices

View full text Add to dashboard Cite

show abstract

“…which form the so called primitive basis. An interesting new line of research was opened by Abardia-Wannerer [2], who studied versions of the classical isoperimetric inequality, but with the usual intrinsic volumes replaced by unitarily invariant valuations. In small degrees (k ≤ 3) they study which linear combinations of hermitian intrinsic volumes satisfy an Alexandrov-Fenchel-type inequality, from which several other isoperimetric inequalities can be derived.…”

Section: Vector Space Structurementioning

confidence: 99%

Valuations and Curvature Measures on Complex Spaces

Bernig

2017

Lecture Notes in Mathematics

View full text Add to dashboard Cite

We survey recent results in hermitian integral geometry, i.e. integral geometry on complex vector spaces and complex space forms. We study valuations and curvature measures on complex space forms and describe how the global and local kinematic formulas on such spaces were recently obtained. While the local and global kinematic formulas in the Euclidean case are formally identical, the local formulas in the hermitian case contain strictly more information than the global ones. Even if one is only interested in the flat hermitian case, i.e. C n , it is necessary to study the family of all complex space forms, indexed by the holomorphic curvature 4λ , and the dependence of the formulas on the parameter λ . We will also describe Wannerer's recent proof of local additive kinematic formulas for unitarily invariant area measures.

show abstract

“…More recently -see [5], [13], [26] for recent surveys -a rapid progress in integral geometry followed Alesker's solution of McMullen's conjecture [2], allowing also to relate the theory to past contributions by other geometers, such as Weyl's tube formula, Chern's kinematic formulas and more, as well to the other, Gelfand-style branch of integral geometry, studying Radon transforms and their generalizations, see [4]. Valuation theory has also been recently used to obtain new types of Brunn-Minkowski [10] and Alexandrov-Fenchel [1] inequalities.…”

mentioning

confidence: 99%

“…This research was partially supported by an NSERC Discovery grant. 1 The classical Crofton formula in its simplest form computes the length of a rectifiable compact curve γ ⊂ R 2 :…”

mentioning

confidence: 99%

Crofton formulas and indefinite signature

Faifman

2017

Geom. Funct. Anal.

View full text Add to dashboard Cite

Abstract. We study the O(p, q)-invariant valuations classified by A. Bernig and the author. Our main result is that every such valuation is given by an O(p, q)-invariant Crofton formula. This is achieved by first obtaining a handful of explicit formulas for a few sufficiently general signatures and degrees of homogeneity, notably in the (p − 1) homogeneous case of O(p, p), yielding a Crofton formula for the centro-affine surface area when p ≡ 3 mod 4. We then exploit the functorial properties of Crofton formulas to pass to the general case. We also identify the invariant formulas explicitly for all O(p, 2)-invariant valuations. The proof relies on the exact computation of some integrals of independent interest. Those are related to Selberg's integral and to the Beta function of a matrix argument, except that the positive-definite matrices are replaced with matrices of all signatures. We also analyze the distinguished invariant Crofton distribution supported on the minimal orbit, and show that, somewhat surprisingly, it sometimes defines the trivial valuation, thus producing a distribution in the kernel of the cosine transform of particularly small support. In the heart of the paper lies the description by Muro of the | det X| s family of distributions on the space of symmetric matrices, which we use to construct a family of O(p, q)-invariant Crofton distributions. We conjecture there are no others, which we then prove for O(p, 2) with p even. The functorial properties of Crofton distributions, which serve an important tool in our investigation, are studied by T. Wannerer and the author in the Appendix.

show abstract

Aleksandrov-Fenchel inequalities for unitary valuations of degree $$2$$ 2 and $$3$$ 3

Abstract: Abstract. We extend the classical Aleksandrov-Fenchel inequality for mixed volumes to functionals arising naturally in hermitian integral geometry. As a consequence, we obtain Brunn-Minkowski and isoperimetric inequalities for hermitian quermassintegrals.

Cited by 13 publications

References 54 publications

Valuations on Convex Functions

Valuations on Convex Functions

Valuations and Curvature Measures on Complex Spaces

Crofton formulas and indefinite signature

Contact Info

Product

Resources

About