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Surjective Hausdorff Isometries of Hyperspaces of Bounded Closed Convex Sets
Abstract
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)→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→ Y defined by F({x})={T(x)} is a surjective affine isometry satisfying F(A)=T[A]:={T(x): x∈ A}, (A∈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.
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BibTeX
@misc{cheng2026surjective,
title = {{Surjective Hausdorff Isometries of Hyperspaces of Bounded Closed Convex Sets}},
author = {Lixin Cheng and Wuyi He and Chulei Liu and Zheming Zheng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18252},
eprint = {2609.18252},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18252},
}
RIS
TY - UNPB
TI - Surjective Hausdorff Isometries of Hyperspaces of Bounded Closed Convex Sets
AU - Lixin Cheng
AU - Wuyi He
AU - Chulei Liu
AU - Zheming Zheng
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18252
UR - https://arxiv.org/abs/2609.18252
AB - 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)→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→ Y defined by F({x})={T(x)} is a surjective affine isometry satisfying F(A)=T[A]:={T(x): x∈ A}, (A∈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.
ER -