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The product of a weak Asplund space and a one-dimensional space is a weak Asplund space: over 45 years of open problem solved

Shaoqiang Shang

TL;DR

The paper resolves the 45-year open question of whether the product of a weak Asplund space with a one-dimensional space preserves the weak Asplund property by developing a framework based on Banach-Mazur game theory, maximal monotone operators, and Minkowski-functionals to relate differentiability in the original and product spaces via projections. The main result shows that if $X$ is weak Asplund, then $X\times \mathbb{R}$ is weak Asplund, with a detailed construction guaranteeing a dense $G_\delta$ set of Gateaux-differentiability points for every continuous convex function on open convex subsets of $X\times \mathbb{R}$. The proof also indicates that the property is stable under taking products with finite-dimensional spaces, thereby strengthening the stability theory of weak Asplund spaces. The results have potential implications for convex optimization, PDE weak-solution frameworks, and stochastic analysis where product-structured spaces arise.

Abstract

In this paper, authors prove that if $X$ is a weak Asplund space, then the space $X\times R$ is a weak Asplund space. Thus the author definitely answered an open problem raised by D.G. Larman and R.R. Phelps for 45 years ago (J. London. Math. Soc. (2), 20(1979), 115--127). The study constructs a framework for proving the existence of densely differentiable sets of convex functions in product spaces through the analysis of Banach-Mazur game theory, maximal monotone operator properties, and the Gateaux differentiability of Minkowski functionals. By associating the convex function properties of the original space and product space via projection mappings, and utilizing sequences of dense open cones to construct $G_δ$-dense subsets, the research ultimately demonstrates that the product space is a weak Asplund space. This work not only enriches the stability theory of weak Asplund spaces and their products with one-dimensional spaces but also provides crucial theoretical support for applications in convex optimization, weak solution construction for partial differential equations, and stochastic analysis.

The product of a weak Asplund space and a one-dimensional space is a weak Asplund space: over 45 years of open problem solved

TL;DR

The paper resolves the 45-year open question of whether the product of a weak Asplund space with a one-dimensional space preserves the weak Asplund property by developing a framework based on Banach-Mazur game theory, maximal monotone operators, and Minkowski-functionals to relate differentiability in the original and product spaces via projections. The main result shows that if is weak Asplund, then is weak Asplund, with a detailed construction guaranteeing a dense set of Gateaux-differentiability points for every continuous convex function on open convex subsets of . The proof also indicates that the property is stable under taking products with finite-dimensional spaces, thereby strengthening the stability theory of weak Asplund spaces. The results have potential implications for convex optimization, PDE weak-solution frameworks, and stochastic analysis where product-structured spaces arise.

Abstract

In this paper, authors prove that if is a weak Asplund space, then the space is a weak Asplund space. Thus the author definitely answered an open problem raised by D.G. Larman and R.R. Phelps for 45 years ago (J. London. Math. Soc. (2), 20(1979), 115--127). The study constructs a framework for proving the existence of densely differentiable sets of convex functions in product spaces through the analysis of Banach-Mazur game theory, maximal monotone operator properties, and the Gateaux differentiability of Minkowski functionals. By associating the convex function properties of the original space and product space via projection mappings, and utilizing sequences of dense open cones to construct -dense subsets, the research ultimately demonstrates that the product space is a weak Asplund space. This work not only enriches the stability theory of weak Asplund spaces and their products with one-dimensional spaces but also provides crucial theoretical support for applications in convex optimization, weak solution construction for partial differential equations, and stochastic analysis.

Paper Structure

This paper contains 3 sections, 9 theorems, 312 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Main Theorems
  3. Some problems

Key Result

Lemma 1.8

(see [12]) Let $T: X\rightarrow2^{X^{*}}$ be a maximal monotone operator and $D=\mathrm{int} D(T)= \mathrm{int} \{x\in X: T(x)\neq \emptyset \}$ is nonempty. Then (1) for every $x\in{D}$, $f_{x,T}(e)=\sigma_{T}\left(x,e\right)$ is subadditive and positive homogeneous and for every $\lambda>0$, we ha (3) the set $(eT)x$ is a singleton if and only if $\sigma_{T}(x,-e)=-\sigma_{T}(x,e)$. (4) if $x_{0

Theorems & Definitions (21)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.6
  • Definition 1.7
  • Lemma 1.8
  • Definition 1.9
  • Lemma 1.10
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • ...and 11 more