Choquet Theory Research Articles

Applications of Strassen’s theorem and Choquet theory to optimal transport problems, to uniformly convex functions and to uniformly smooth functions

We provide a unifying interpretation of various optimal transport problems as a minimisation of a linear functional over the set of all Choquet representations of a given pair of probability measures ordered with respect to a certain convex cone of functions. This allows us to provide novel proofs of duality formulae. Among our tools is Strassen’s theorem.We provide new formulations of the primal and the dual problem for martingale optimal transport employing a novel representation of the set of extreme points of probability measures in convex order on Euclidean space. We exhibit a link to uniformly convex and uniformly smooth functions and provide a new characterisation of such functions.We introduce a notion of martingale triangle inequality. We show that Kantorovich–Rubinstein duality bears an analogy in the martingale setting employing the cost functions that satisfy the inequality.

Assemblages and steering in general probabilistic theories

We study steering in the framework of general probabilistic theories. We show that for dichotomic assemblages, steering can be characterized in terms of a certain tensor cross norm, which is also related to a steering degree given by steering robustness. Another contribution is the observation that steering in general probabilistic theories (GPTs) can be conveniently treated using Choquet theory for probability measures on the state space. In particular, we find a variational expression for universal steering degree for dichotomic assemblages and obtain conditions characterizing unsteerable states analogous to some conditions recently found for the quantum case. The setting also enables us to rather easily extend the results to infinite dimensions and arbitrary numbers of measurements with arbitrary outcomes.

Complex Geodesics in Tube Domains and Their Role in the Study of Harmonic Mappings in the Disc

We continue the research on the structure of complex geodesics in tube domains over (bounded) convex bases. In some special cases a more explicit form of the geodesics than the existing ones are provided. As one of the consequences of our study an effective formula for the Kobayashi—Royden metric in the tube domain \({T_{{\mathbb{B}_n}}}\) at the origin is given. The results on the Kobayashi—Royden metric in a natural way provide versions of the Schwarz Lemma for harmonic mappings. We also present a result on harmonic mappings defined on the disc that may be seen as a generalisation of the Radó—Kneser—Choquet Theorem for a class of harmonic bivalent mappings that lets understand better the geometry of complex geodesics in tube domains.

On Hahn-Banach theorem and some of its applications

Abstract First, this work provides an overview of some of the Hahn-Banach type theorems. Of note, some of these extension results for linear operators found recent applications to isotonicity of convex operators on a convex cone. Next, the work investigates applications of the Krein-Milman theorem to representation theory and elements of Choquet theory. A sandwich theorem of intercalating an affine function h h between f f and g , g, where f f\hspace{.25em} and – g \mbox{--}g are convex, f ≤ g f\le g on a finite-simplicial set, is recalled. Its recent topological version is also noted. Here, the novelty is that a finite-simplicial set may be unbounded in any locally convex topology on the domain space. Third, the paper summarizes and comments on recently published applications of a Hahn-Banach extension result for positive linear operators, combined with polynomial approximation on unbounded subsets, to the Markov moment problem. Some applications to concrete spaces are detailed as well. Finally, this work provides a characterization of a finite-dimensional convex bounded subset in terms of the property that any convex function defined on that subset is bounded below. This last property remains valid for a large class of convex operators.

A note on the Choquet type operators

In this note the Choquet type operators are introduced, in connection to Choquet's theory of integrability with respect to a not necessarily additive set function. Based on their properties, a quantitative estimate for the nonlinear Korovkin type approximation theorem associated to Bernstein-Kantorovich-Choquet operators is proved. The paper also includes a large generalization of H\"{o}lder's inequality within the framework of monotone and sublinear operators acting on spaces of continuous functions.

Francesco Altomare - the remarkable mathematician and human being

We laconically describe the great contributions of Professor Francesco Altomare to mathematical research and PhD education, and his unique status in the mathematical community. In particular, we present and give examples of his innovative and great achievements related to the following areas of mathematics: Functional Analysis, Operator Theory, Potential Theory, Approximation Theory, Probability Theory, Function Spaces, Choquet's Theory, Dirichlet's Problem and Semigroup Theory. 
 Moreover, we report on and give concrete examples of his unique way to work together with PhD students, both before and sometimes also after their dissertation. Finally, we shortly describe his remarkable “class travel” from “simple” conditions with no academic traditions in his family in the small hometown Giovinazzo to finally become the broad, ingenious, and powerful mathematician he is regarded to be today.

Robert R. Phelps: Lectures on Choquet's Theorem. 2nd ed., Lecture Notes in Math., 1757, Springer, 2001年，viii+124ページ．

Robert R. Phelps: Lectures on Choquet's Theorem. 2nd ed., Lecture Notes in Math., 1757, Springer, 2001年，viii+124ページ．

The Radó–Kneser–Choquet theorem for $p$-harmonic mappings between Riemannian surfaces

In the planar setting, the Radó–Kneser–Choquet theorem states that a harmonic map from the unit disk onto a Jordan domain bounded by a convex curve is a diffeomorphism provided that the boundary mapping is a homeomorphism. We prove the injectivity criterion of Radó–Kneser–Choquet for p -harmonic mappings between Riemannian surfaces. In our proof of the injectivity criterion we approximate the p -harmonic map with auxiliary mappings that solve uniformly elliptic systems. We prove that each auxiliary mapping has a positive Jacobian by a homotopy argument. We keep the maps injective all the way through the homotopy with the help of the minimum principle for a certain subharmonic expression that is related to the Jacobian.

Robust expected utility maximization with medial limits

In this paper we study a robust expected utility maximization problem with random endowment in discrete time. We give conditions under which an optimal strategy exists and derive a dual representation for the optimal utility. Our approach is based on a general representation result for monotone convex functionals, a functional version of Choquet's capacitability theorem and medial limits. The novelty is that it works under nondominated model uncertainty without any assumptions of time-consistency. As applications, we discuss robust utility maximization problems with moment constraints, Wasserstein constraints and Wasserstein penalties.

Schoenberg’s theorem for real and complex Hilbert spheres revisited

Schoenberg’s theorem for the complex Hilbert sphere proved by Christensen and Ressel in 1982 by Choquet theory is extended to the following result: Let L denote a locally compact group and let D¯ denote the closed unit disc in the complex plane. Continuous functions f:D¯×L→C such that f(ξ⋅η,u−1v) is a positive definite kernel on the product of the unit sphere in ℓ2(C) and L are characterized as the functions with a uniformly convergent expansion f(z,u)=∑m,n=0∞φm,n(u)zmz¯n,where φm,n is a double sequence of continuous positive definite functions on L such that ∑φm,n(eL)<∞ (eL is the neutral element of L). It is shown how the coefficient functions φm,n are obtained as limits from expansions for positive definite functions on finite dimensional complex spheres via a Rodrigues formula for disc polynomials.Similar results are obtained for the real Hilbert sphere.

Outer measure on effect algebras

Abstract In the present paper, the notion of an outer measure m * on lattice ordered effect algebras L is introduced and investigated. Carathéodory’s criterion for the outer measure m * is given and established on L. Properties of an induced outer measure m * defined on a lattice ordered effect algebra L are studied and finally, Choquet theorem is proved for the induced outer measure m * on a σ-complete lattice ordered effect algebra L.

ON LOCAL HUDETZ g-ENTROPY

In this paper, a local approach to the concept of Hudetz $g$-entropy is presented. The introduced concept is stated in terms of Hudetz $g$-entropy. This representation is based on the concept of $g$-ergodic decomposition which is a result of the Choquet's representation Theorem for compact convex metrizable subsets of locally convex spaces.

Radó–Kneser–Choquet theorem for harmonic mappings between surfaces

We simplify and improve a recent result of Martin (Trans AMS 368:647–658, 2016). Then we prove that if f is an orientation preserving harmonic mapping of the unit disk onto a $$C^{3,\alpha }$$ surface $$\Sigma $$ bounded by a Jordan curve $$\gamma \in C^{3,\alpha }$$ , that belongs to the boundary of a convex domain in $$\mathbf {R}^3$$ , then f is a diffeomorphism.

Invertible harmonic mappings of the unit disk onto Dini smooth Jordan domains

In this paper we extend Radó–Kneser–Choquet theorem for the mappings with weak homeomorphic Lipschitz boundary function and Dini's smooth boundary but without restriction on the convexity of the image domain, provided that the Jacobian satisfies a certain boundary condition. The proof is based on a recent extension of Radó–Kneser–Choquet theorem by Alessandrini and Nesi [1] and is used the approximation principle.

A representation theorem for standard weighted harmonic mappings with an integer exponent and its applications

In this paper, we study α¯-harmonic mappings with an integer exponent and obtain a representation theorem which determines the relation between α¯-harmonic mappings and Euclidean harmonic mappings. As two applications of this representation theorem, we obtain a counterexample of the Radó–Kneser–Choquet theorem for α¯-harmonic mappings and show that the Lipschitz continuity of an α¯-harmonic mapping with an integer exponent is determined by the Euclidean harmonic mapping with same boundary.

On Liouville-type theorems and the uniqueness of the positive Cauchy problem for a class of hypoelliptic operators

This note contains a representation formula for positive solutions of linear degenerate second-order equations of the form $$ \partial_t u (x,t) = \sum_{j=1}^m X_j^2 u(x,t) + X_0 u(x,t) \qquad (x,t) \in \mathbb{R}^N \times\, ]- \infty ,T[,$$ proved by a functional analytic approach based on Choquet theory. As a consequence, we obtain Liouville-type theorems and uniqueness results for the positive Cauchy problem.

Orthogonal Measures on State Spaces and Context Structure of Quantum Theory

An interplay between recent topos theoretic approach and standard convex theoretic approach to quantum theory is discovered. Combining new results on isomorphisms of posets of all abelian subalgebras of von Neumann algebras with classical Tomita’s theorem from state space Choquet theory, we show that order isomorphisms between the sets of orthogonal measures (resp. finitely supported orthogonal measures) on state spaces endowed with the Choquet order are given by Jordan ∗-isomorphims between corresponding operator algebras. It provides new complete Jordan invariants for σ-finite von Neumann algebras in terms of decompositions of states and shows that one can recover physical system from associated structure of convex decompositions (discrete or continuous) of a fixed state.

Generalized Gauss inequalities via semidefinite programming

A sharp upper bound on the probability of a random vector falling outside a polytope, based solely on the first and second moments of its distribution, can be computed efficiently using semidefinite programming. However, this Chebyshev-type bound tends to be overly conservative since it is determined by a discrete worst-case distribution. In this paper we obtain a less pessimistic Gauss-type bound by imposing the additional requirement that the random vector’s distribution must be unimodal. We prove that this generalized Gauss bound still admits an exact and tractable semidefinite representation. Moreover, we demonstrate that both the Chebyshev and Gauss bounds can be obtained within a unified framework using a generalized notion of unimodality. We also offer new perspectives on the computational solution of generalized moment problems, since we use concepts from Choquet theory instead of traditional duality arguments to derive semidefinite representations for worst-case probability bounds.

Characterizing joint distributions of random sets by multivariate capacities

By the Choquet theorem, distributions of random closed sets can be characterized by a certain class of set functions called capacity functionals. In this paper a generalization to the multivariate case is presented, that is, it is proved that the joint distribution of finitely many random sets can be characterized by a multivariate set function being completely alternating in each component, or alternatively, by a capacity functional defined on complements of cylindrical sets. For the special case of finite spaces a multivariate version of the Moebius inversion formula is derived. Furthermore, we use this result to formulate an existence theorem for set-valued stochastic processes.

The singularity of positive linear functionals

Our purpose is to show that the various concepts of singularity of representable positive functionals on ∗-algebras coincide, moreover to present a new characterization of singularity by means of Choquet theory of the state space. In the context of singularity, the paper includes an equivalent condition for a representable positive functional to be pure.