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Unitarily invariant valuations on convex functions

Jonas Knoerr

TL;DR

This work extends valuation theory to unitarily invariant valuations on convex functions, linking geometric, analytic, and algebraic structures. It shows that smooth, unitary-invariant valuations decompose into a direct sum controlled by Monge-Ampère-type operators, and that homogeneous valuations are determined by restrictions to extremal subspaces, a fact encoded in the Goodey-Weil distribution’s Fourier-Laplace transform. The authors construct invariant differential forms, derive explicit integral representations, and develop a Fourier-analytic framework using invariant polynomials and equivariant operators to express the GW-transform. A key achievement is a precise decomposition of GW$(\mu)$ into contributions from the $\Theta^n_{k,q}$ and $\Upsilon^n_{k,q}$ families, and a general Fourier decomposition formula for arbitrary $\mathrm{U}(n)$-invariant valuations, which reduces to the smooth case and yields a density/uniqueness mechanism via subspace restrictions. Together, these results advance the classification of $\mathrm{VConv}_k(\mathbb{C}^n)^{\mathrm{U}(n)}$ valuations and set the stage for a complete two-part classification.

Abstract

Continuous, dually epi-translation invariant valuations on the space of finite-valued convex functions on $\mathbb{C}^n$ that are invariant under the unitary group are investigated. It is shown that elements belonging to the dense subspace of smooth valuations admit a unique integral representation in terms of two families of Monge-Ampère-type operators. In addition, it is proved that homogeneous valuations are uniquely determined by restrictions to subspaces of appropriate dimension and that this information is encoded in the Fourier-Laplace transform of the associated Goodey-Weil distributions. These results are then used to show that a continuous unitarily invariant valuation is uniquely determined by its restriction to a certain finite family of subspaces of $\mathbb{C}^n$.

Unitarily invariant valuations on convex functions

TL;DR

This work extends valuation theory to unitarily invariant valuations on convex functions, linking geometric, analytic, and algebraic structures. It shows that smooth, unitary-invariant valuations decompose into a direct sum controlled by Monge-Ampère-type operators, and that homogeneous valuations are determined by restrictions to extremal subspaces, a fact encoded in the Goodey-Weil distribution’s Fourier-Laplace transform. The authors construct invariant differential forms, derive explicit integral representations, and develop a Fourier-analytic framework using invariant polynomials and equivariant operators to express the GW-transform. A key achievement is a precise decomposition of GW into contributions from the and families, and a general Fourier decomposition formula for arbitrary -invariant valuations, which reduces to the smooth case and yields a density/uniqueness mechanism via subspace restrictions. Together, these results advance the classification of valuations and set the stage for a complete two-part classification.

Abstract

Continuous, dually epi-translation invariant valuations on the space of finite-valued convex functions on that are invariant under the unitary group are investigated. It is shown that elements belonging to the dense subspace of smooth valuations admit a unique integral representation in terms of two families of Monge-Ampère-type operators. In addition, it is proved that homogeneous valuations are uniquely determined by restrictions to subspaces of appropriate dimension and that this information is encoded in the Fourier-Laplace transform of the associated Goodey-Weil distributions. These results are then used to show that a continuous unitarily invariant valuation is uniquely determined by its restriction to a certain finite family of subspaces of .
Paper Structure (17 sections, 51 theorems, 191 equations)

This paper contains 17 sections, 51 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Plan of the article
  4. The Fourier-Laplace transform of Goodey-Weil distributions
  5. Basic properties and relation to restrictions to subspaces
  6. Restrictions on the support
  7. Unitarily invariant differential forms and smooth valuations
  8. The algebra $[\Lambda^*(\mathbb{C}^n\times\mathbb{C}^n)^*]^{\mathrm{U}(n)}$
  9. Invariant differential forms
  10. A pointwise relation
  11. Representation of smooth valuations
  12. The Fourier-Laplace transform for $\mathrm{U}(n)$-invariant valuations
  13. A family of invariant polynomials
  14. Two families of equivariant differential operators
  15. The Fourier-Laplace transform in the smooth case
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $G\subset \mathrm{SO}(n)$ be a compact subgroup. The space $\mathop{\mathrm{Val}}\nolimits(\mathbb{R}^n)^{G}$ of all continuous, translation and $G$-invariant valuations on $\mathcal{K}(\mathbb{R}^n)$ is finite dimensional if and only if $G$ operates transitively on the unit sphere.

Theorems & Definitions (91)

  • Theorem 1.1: Alesker AleskerP.McMullensconjecture2000 Theorem 8.1
  • Theorem 1.2: Bernig-Fu BernigFuHermitianintegralgeometry2011 Theorem 3.2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 2.1: Knoerrsupportduallyepi2021 Theorem 2
  • Proposition 2.2: Knoerrsupportduallyepi2021 Proposition 6.3
  • Corollary 2.3
  • proof
  • ...and 81 more