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Mazur-Ulam Theorem With Gromov-Hausdorff Distance

S. A. Bogaty, A. A. Tuzhilin

Abstract

It is shown that two Banach spaces are linearly isometric if and only if the Gromov--Hausdorff distance between them is finite, in particular, zero. The proof is compilative and relies on results obtained by many researchers on the approximability of almost-surjective almost-isometries by linear surjective isometries. In the finite-dimensional case, previously obtained by I.~Mikhailov, a simpler proof under weaker assumptions is given. In the finite-dimensional case, a criterion for isometry in terms of finite (compact) subsets is also given.

Mazur-Ulam Theorem With Gromov-Hausdorff Distance

Abstract

It is shown that two Banach spaces are linearly isometric if and only if the Gromov--Hausdorff distance between them is finite, in particular, zero. The proof is compilative and relies on results obtained by many researchers on the approximability of almost-surjective almost-isometries by linear surjective isometries. In the finite-dimensional case, previously obtained by I.~Mikhailov, a simpler proof under weaker assumptions is given. In the finite-dimensional case, a criterion for isometry in terms of finite (compact) subsets is also given.
Paper Structure (17 theorems, 18 equations)

This paper contains 17 theorems, 18 equations.

Key Result

Theorem 1

For compact Hausdorff spaces $X$ and $Y$, the following conditions are equiva-lent :

Theorems & Definitions (32)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof
  • Corollary 4
  • Remark 5
  • Proposition 6
  • proof
  • Theorem 7
  • proof
  • ...and 22 more