[Paper Review] Valuations on convex functions and convex sets and Monge-Ampere operators

Semyon Alesker|arXiv (Cornell University)|Mar 26, 2017

Geometry and complex manifolds4 citations

TL;DR

This paper establishes a deep connection between continuous valuations on convex functions invariant under addition of linear functionals and translation-invariant continuous valuations on convex bodies. Using real, complex, quaternionic, and octonionic Monge-Ampère operators, it constructs a linear map from the former space to the latter, proving the image is dense and the kernel is infinite-dimensional, thereby unifying higher-dimensional geometric valuation theory across division algebras.

ABSTRACT

The notion of a valuation on convex bodies is very classical. The notion of a valuation on a class of functions was recently introduced and studied by M. Ludwig and others. We study an explicit relation between continuous valuations on convex functions which are invariant under adding arbitrary linear functionals, and translations invariant continuous valuations on convex bodies. More precisely, we construct a natural linear map from the former space to the latter and prove that it has dense image and infinite dimensional kernel. The proof uses the author's irreducibility theorem and few properties of the real Monge-Ampere operators due to A.D. Alexandrov and Z. Blocki. Fur- thermore we show how to use complex, quaternionic, and octonionic Monge-Ampere operators to construct more examples of continuous valuations on convex functions in an analogous way.

Motivation & Objective

• To explore the structural relationship between valuations on convex functions and valuations on convex bodies.
• To extend the theory of valuations beyond classical convex sets to include convex functions with invariance under linear functionals.
• To demonstrate that Monge-Ampère operators in multiple algebraic settings (real, complex, quaternionic, octonionic) generate new classes of continuous valuations on convex functions.
• To prove that the induced map from function-valuations to body-valuations has dense image and infinite-dimensional kernel, thus revealing a rich algebraic structure.

Proposed method

• Define a linear map $ T: VConv(V^*) \to Val(V) $ that sends a valuation $ \Phi $ on convex functions on the dual space $ V^* $ to a valuation $ \phi $ on convex bodies in $ V $ via $ \phi(K) = \Phi(h_K) $, where $ h_K $ is the support function of $ K $.
• Prove that $ T $ commutes with the action of $ GL(V) $, enabling the use of the author’s irreducibility theorem to analyze the image of $ T $.
• Use the real Monge-Ampère operator (in the sense of Alexandrov and Blocki) to construct explicit examples of continuous valuations on convex functions that are invariant under addition of linear functionals.
• Generalize the construction to complex, quaternionic, and octonionic Monge-Ampère operators, leveraging recent results on octonionic Hessian and determinant for two variables.
• Show that the resulting functionals satisfy the valuation property and continuity in the $ C^0 $-topology, placing them in $ VConv(V^*) $.
• Apply the irreducibility theorem to show that the image of $ T $ intersects every irreducible $ GL(V) $-subrepresentation of $ Val(V) $, implying density.

Experimental results

Research questions

• RQ1 How are continuous valuations on convex functions invariant under addition of linear functionals related to translation-invariant valuations on convex bodies?
• RQ2 Can Monge-Ampère operators in non-commutative and non-associative algebras (like quaternions and octonions) generate new classes of valuations on convex functions?
• RQ3 What is the image of the natural map $ T: VConv(V^*) \to Val(V) $, and how does it relate to the full space of translation-invariant valuations on convex bodies?
• RQ4 To what extent do the kernels of such maps reflect the algebraic complexity of the underlying function spaces?
• RQ5 Can higher-dimensional analogues of Kazarnovskii’s pseudovolume be constructed using octonionic Monge-Ampère operators?

Key findings

• The linear map $ T: VConv(V^*) \to Val(V) $ has a dense image in the space of continuous translation-invariant valuations on convex bodies.
• The kernel of $ T $ is infinite-dimensional, indicating a vast space of function-valued valuations that map to the zero functional on convex bodies.
• The construction of valuations on convex functions using the real Monge-Ampère operator yields functionals that are continuous and satisfy the valuation property, as established by Blocki and Alexandrov.
• Analogous constructions using complex, quaternionic, and octonionic Monge-Ampère operators produce continuous valuations on convex functions invariant under addition of linear functionals.
• For octonionic functions of two variables, the mixed determinant of the octonionic Hessian defines a signed measure that gives rise to continuous valuations via integration against compactly supported functions.
• When restricted to support functions, the resulting valuations on convex bodies are proportional to each other, forming an octonionic analogue of Kazarnovskii’s pseudovolume.