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Superforms, supercurrents and convex geometry

Bo Berndtsson

TL;DR

The paper develops a real-analytic framework based on Lagerberg's superforms to study convex geometry, unifying valuations, Monge–Ampère theory, and mixed volumes. It constructs Bedford–Taylor-type wedge products for convex potentials, defines Monge–Ampère measures $MA(\psi)$, and proves key identities like $MA(h_K)=|K|$, linking geometric volume to Monge–Ampère masses. It then treats valuations on convex bodies via wedge powers of $dd^{\#}h_K$, establishing a Fourier-transform–like correspondence $\mathcal{F}$ between currents and valuations, and analyzes smooth and monotone valuations through strong, strongly homogeneous currents and the normal cycle. Finally, it provides two independent proofs of the Alexandrov–Fenchel inequalities—via Alexandrov's operator on the sphere and via current-based, toric-free arguments—thereby connecting convex-geometric inequalities to a broader Kähler-like calculus on real manifolds.

Abstract

We develop the calculus of superforms as a tool for convex geometry. The formalism is applied to valuations on convex bodies, the Alexandrov-Fenchel inequalities and Monge- Ampère equations on the boundary of convex bodies.

Superforms, supercurrents and convex geometry

TL;DR

The paper develops a real-analytic framework based on Lagerberg's superforms to study convex geometry, unifying valuations, Monge–Ampère theory, and mixed volumes. It constructs Bedford–Taylor-type wedge products for convex potentials, defines Monge–Ampère measures , and proves key identities like , linking geometric volume to Monge–Ampère masses. It then treats valuations on convex bodies via wedge powers of , establishing a Fourier-transform–like correspondence between currents and valuations, and analyzes smooth and monotone valuations through strong, strongly homogeneous currents and the normal cycle. Finally, it provides two independent proofs of the Alexandrov–Fenchel inequalities—via Alexandrov's operator on the sphere and via current-based, toric-free arguments—thereby connecting convex-geometric inequalities to a broader Kähler-like calculus on real manifolds.

Abstract

We develop the calculus of superforms as a tool for convex geometry. The formalism is applied to valuations on convex bodies, the Alexandrov-Fenchel inequalities and Monge- Ampère equations on the boundary of convex bodies.
Paper Structure (16 sections, 53 theorems, 479 equations)

This paper contains 16 sections, 53 theorems, 479 equations.

Table of Contents

  1. Introduction
  2. Brief recap of superforms and some conventions.
  3. Positivity of superforms
  4. The theory of Bedford and Taylor.
  5. Volumes and mixed volumes of convex bodies.
  6. Homogeneous forms.
  7. Fourier transforms
  8. Charts.
  9. Valuations on convex bodies.
  10. The normal cycle
  11. Characterization of strong forms
  12. Alexandrov's differential operator.
  13. Two proofs of the Alexandrov-Fenchel inequalities.
  14. Alexandrov's proof
  15. A real variable variant of Gromov's proof.
  16. ...and 1 more sections

Key Result

Theorem 3.1

A closed $(p,p)$-form (thus in particular a form with constant coefficients), $\Omega$, that satisfies $T(\Omega)=0$ is strong. In other words, $S=Ker(T)$.

Theorems & Definitions (107)

  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • proof
  • Lemma 3.6
  • ...and 97 more