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Convex valuations, from Whitney to Nash

Dmitry Faifman, Georg C. Hofstätter

TL;DR

The paper solves Whitney-type extension problems for smooth valuations by combining Alesker–Fourier duality, double-forms, and compatibility on intersections. It proves that, for the full Grassmannian $S=\mathrm{Gr}_r(\mathbb{R}^n)$ with $r\ge j+2$, compatibility is sufficient to extend a given $j$-homogeneous valuation from all $E\in S$, and it provides a constructive mechanism via extension of double forms. Building on this, it develops extension results for finite subspace arrangements and for compact submanifolds, leading to a Nash-type embedding theorem for valuations on compact manifolds and to Crofton formulas for all smooth valuations on manifolds, including explicit Crofton formulas for odd translation-invariant valuations. The work unifies extension problems across linear spaces and manifolds, yielding new existence results in valuation theory and broad Crofton-type representations. It also clarifies the structure of the cosine transform’s image and demonstrates the localization of density-to-valuation transition to codimension two, with substantial implications for integral geometry on manifolds.

Abstract

We consider the Whitney problem for valuations: does a smooth $j$-homogeneous translation-invariant valuation on $\mathbb R^n$ exist that has given restrictions to a fixed family $S$ of linear subspaces? A necessary condition is compatibility: the given valuations must coincide on intersections. We show that for $S=\mathrm{Gr}_r(\mathbb R^n)$, the grassmannian of $r$-planes, this condition becomes sufficient once $r\geq j+2$. This complements the Klain and Schneider uniqueness theorems with an existence statement, and provides a recursive description of the image of the cosine transform. Informally speaking, we show that the transition from densities to valuations is localized to codimension $2$. We then look for conditions on $S$ when compatibility is also sufficient for extensibility, in two distinct regimes: finite arrangements of subspaces, and compact submanifolds of the grassmannian. In both regimes we find unexpected flexibility. As a consequence of the submanifold regime, we prove a Nash-type theorem for valuations on compact manifolds, from which in turn we deduce the existence of Crofton formulas for all smooth valuations on manifolds. As an intermediate step of independent interest, we construct Crofton formulas for all odd translation-invariant valuations.

Convex valuations, from Whitney to Nash

TL;DR

The paper solves Whitney-type extension problems for smooth valuations by combining Alesker–Fourier duality, double-forms, and compatibility on intersections. It proves that, for the full Grassmannian with , compatibility is sufficient to extend a given -homogeneous valuation from all , and it provides a constructive mechanism via extension of double forms. Building on this, it develops extension results for finite subspace arrangements and for compact submanifolds, leading to a Nash-type embedding theorem for valuations on compact manifolds and to Crofton formulas for all smooth valuations on manifolds, including explicit Crofton formulas for odd translation-invariant valuations. The work unifies extension problems across linear spaces and manifolds, yielding new existence results in valuation theory and broad Crofton-type representations. It also clarifies the structure of the cosine transform’s image and demonstrates the localization of density-to-valuation transition to codimension two, with substantial implications for integral geometry on manifolds.

Abstract

We consider the Whitney problem for valuations: does a smooth -homogeneous translation-invariant valuation on exist that has given restrictions to a fixed family of linear subspaces? A necessary condition is compatibility: the given valuations must coincide on intersections. We show that for , the grassmannian of -planes, this condition becomes sufficient once . This complements the Klain and Schneider uniqueness theorems with an existence statement, and provides a recursive description of the image of the cosine transform. Informally speaking, we show that the transition from densities to valuations is localized to codimension . We then look for conditions on when compatibility is also sufficient for extensibility, in two distinct regimes: finite arrangements of subspaces, and compact submanifolds of the grassmannian. In both regimes we find unexpected flexibility. As a consequence of the submanifold regime, we prove a Nash-type theorem for valuations on compact manifolds, from which in turn we deduce the existence of Crofton formulas for all smooth valuations on manifolds. As an intermediate step of independent interest, we construct Crofton formulas for all odd translation-invariant valuations.
Paper Structure (20 sections, 59 theorems, 180 equations)

This paper contains 20 sections, 59 theorems, 180 equations.

Table of Contents

  1. Introduction and results
  2. Background and preparations
  3. Double forms
  4. Valuations
  5. Smooth valuations and differential forms
  6. Klain and Schneider embedding
  7. Operations on valuations
  8. Extension from the full grassmannian
  9. Extending double forms
  10. Smoothness of compatible sections
  11. Proof of Theorem \ref{['mthm:resProperty']}
  12. Extension from finite arrangements of subspaces
  13. Preparations and extensions of linear forms
  14. Proof of Theorem \ref{['mthm:extensionFinArrang']}
  15. Extension from compact submanifolds and the Nash theorem
  16. ...and 5 more sections

Key Result

Theorem 1

Suppose that $M$ is a compact smooth manifold. Then there exists an embedding $e:M\hookrightarrow \mathbb{R}^N$ such that $e^*\left(\mathop{\mathrm{Val}}\nolimits^\infty(\mathbb{R}^N)\right)=\mathcal{W}_1^\infty(M)\oplus \mathop{\mathrm{Span}}\nolimits\{\chi\}$.

Theorems & Definitions (82)

  • Theorem 1
  • Definition 1.1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 1.2: Klain2000Schneider1996
  • Theorem 1.3
  • Corollary 1.4
  • Definition 1.5
  • Theorem 5
  • ...and 72 more