arXiv subjects

Chulei Liu

Publications and source records attributed to Chulei Liu.

3 recordsLinked to original sources

arXiv2026

Surjective Hausdorff Isometries of Hyperspaces of Bounded Closed Convex Sets

Let $X$ and $Y$ be real Banach spaces, and let $\cC(X)$ and $\cC(Y)$ denote the families of all nonempty bounded closed convex subsets of $X$ and $Y$, respectively, equipped with the Hausdorff metric. We prove that every surjective isometry $F:\cC(X)\to\cC(Y)$ is induced by a surjective affine isometry of the underlying spaces. More precisely, $F$ maps singleton sets onto singleton sets, and the map $T:X\to Y$ defined by $F(\{x\})=\{T(x)\}$ is a surjective affine isometry satisfying \[ F(A)=T[A]:=\{T(x):\;x\in A\},\qquad(A\in\cC(X)). \] In particular, our theorem extends the result of Gruber and Lettl for finite-dimensional Euclidean spaces to arbitrary real Banach spaces, without imposing any additional assumptions on the underlying spaces.

math.FA↗

arXiv2026

On strengthened versions of Klee's convex body problem in Banach spaces

In a recent article, Cheng, Jiang and Yuan gave an affirmative answer to Klee's convex bodies problem of Banach spaces in the sense of strict convexity and Gâteaux smoothness. In this paper, we continue to study this problem in strong senses, such as local uniform convexity, uniform convexity, Fréchet smoothness and uniform smoothness. As a result, we show (1) Every convex body in a Banach space $X$ is approximated by locally uniformly convex bodies with respect to the Hausdorff metric if and only if $X$ admits an equivalent locally uniformly convex norm; (2) Every convex body in $X$ can be approximated by Fréchet smooth convex bodies if $X$ admits an equivalent norm so that its dual norm is locally uniformly convex on $X^*$; 3. Every convex body in $X$ can be approximated by both locally uniformly convex and Fréchet smooth convex bodies if $X$ is reflexive; 4. If $X$ is separable, then every convex body in $X$ can be approximated by both locally uniformly convex and Fréchet smooth convex bodies if and only if $X$ is an Asplund space; (5) the following statements are equivalent: A. $X$ is super reflexive; B. Every convex body in $X$ can be uniformly approximated by uniformly convex bodies; C. Every convex body in $X$ can be uniformly approximated by uniformly smooth convex bodies; D. Every convex body in $X$ can be uniformly approximated by both uniformly convex and uniformly smooth convex bodies.

math.FA↗

arXiv2026

An Extension and Refinement of the Brouwer-Schauder-Tychonoff Fixed Point Theorem

In this paper, we present the Brouwer-Schauder-Tychonoff fixed point theorem on locally convex spaces as the following extension and improvement: Suppose that S is a compact star-shaped subset with respect to p in S with its convexity index alpha(p)>0. Then every continuous self-mapping f has one of the following two properties: (a) The point p is a fixed point of f; (b) f has uncountably many different eigenvalues and eigenvectors. Note that a closed bounded star-shaped set in a locally convex space is convex if and only if alpha=1, and we extend a Brouwer's type fixed-point theorem on compact star-shaped sets in Banach spaces in a more concise manner to locally convex spaces, thereby this is a simplification and an improvement of the Tychonoff fixed-point theorem to compact star-shaped sets.

math.FA↗