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On Popoviciu's concept of convexity for functions of $d$ variables

Andrzej Komisarski, Teresa Rajba

TL;DR

The paper develops a multidimensional generalization of Popoviciu's higher-order convexity by introducing box-$\mathbf n$-convex functions for $\mathbf n=(n_1,\dots,n_d)$ and arbitrary open boxes. It derives integral representations that decompose box-$\mathbf n$-convex functions into pseudo-polynomials plus kernel-based measures, enabling differential methods via $\mathbf n$-regularity and leading to Raşa, Jensen, and Hermite-Hadamard inequalities in this setting. It also establishes stochastic box-$\mathbf n$-convex orders, with characterizations in terms of moments and convolutions, and discusses probabilistic representations and uniqueness of the representing measures. Overall, the work unifies and extends prior two-variable differentiable results to the general $d$-dimensional, non-regular regime, providing a robust framework for functional inequalities in higher-order convexity.

Abstract

We establish an integral representation for Popoviciu's convex functions of $d$ variables. This representation serves as a~foundation for deriving several functional inequalities, analogous to those well-known for usual convex functions. Our results generalize and extend the results obtained by S.~Gal, C.~Niculescu, B.~Gavrea, T.~Popoviciu, and others, who considered only differentiable functions of two variables. In contrast to other authors, we do not impose any additional regularity assumptions on the studied functions.

On Popoviciu's concept of convexity for functions of $d$ variables

TL;DR

The paper develops a multidimensional generalization of Popoviciu's higher-order convexity by introducing box--convex functions for and arbitrary open boxes. It derives integral representations that decompose box--convex functions into pseudo-polynomials plus kernel-based measures, enabling differential methods via -regularity and leading to Raşa, Jensen, and Hermite-Hadamard inequalities in this setting. It also establishes stochastic box--convex orders, with characterizations in terms of moments and convolutions, and discusses probabilistic representations and uniqueness of the representing measures. Overall, the work unifies and extends prior two-variable differentiable results to the general -dimensional, non-regular regime, providing a robust framework for functional inequalities in higher-order convexity.

Abstract

We establish an integral representation for Popoviciu's convex functions of variables. This representation serves as a~foundation for deriving several functional inequalities, analogous to those well-known for usual convex functions. Our results generalize and extend the results obtained by S.~Gal, C.~Niculescu, B.~Gavrea, T.~Popoviciu, and others, who considered only differentiable functions of two variables. In contrast to other authors, we do not impose any additional regularity assumptions on the studied functions.
Paper Structure (10 sections, 51 theorems, 135 equations)

This paper contains 10 sections, 51 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Selected properties of divided differences of order $n$
  3. Divided differences of order $(n_1,\ldots, n_d)$ and box-$(n_1,\ldots, n_d)$-convex functions
  4. Pseudo-polynomials and box-$\mathbf n$-affine functions
  5. $\mathbf n$-regular functions
  6. Integration and differentiation of $\mathbf n$-regular functions
  7. Integral representation of box-monotone functions
  8. Integral representation of box-$\textbf{n}$-convex functions
  9. Box-$\mathbf n$-convex orders.
  10. The Hermite-Hadamard, Jensen and Raşa inequalities

Key Result

Proposition 1

A function $f(x)$ is $n$-convex if and only if its derivative $f^{(n-1)}(x)$ exists and is convex (with the convention $f^{(0)}(x)=f(x)$).

Theorems & Definitions (105)

  • Proposition 1: Popoviciu1934
  • Proposition 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Definition 1
  • Proposition 3
  • proof : Sketch of the proof
  • Remark 3
  • ...and 95 more