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Classical and continuous Gromov-Hausdorff distances

K. V. Semenov, A. A. Tuzhilin

Abstract

Starting from the definition of the Gromov-Hausdorff distance via distortion of correspondences, we add the requirement of semicontinuity of each correspondence and its inverse. It turns out that in the case of lower semicontinuity we obtain the same classical Gromov-Hausdorff distance, while for upper semicontinuity we are able to prove coincidence with the classical one only in cases where the spaces are either totally bounded or boundedly compact.

Classical and continuous Gromov-Hausdorff distances

Abstract

Starting from the definition of the Gromov-Hausdorff distance via distortion of correspondences, we add the requirement of semicontinuity of each correspondence and its inverse. It turns out that in the case of lower semicontinuity we obtain the same classical Gromov-Hausdorff distance, while for upper semicontinuity we are able to prove coincidence with the classical one only in cases where the spaces are either totally bounded or boundedly compact.

Paper Structure

This paper contains 5 sections, 28 theorems, 18 equations.

Table of Contents

  1. Basic definitions and preliminary results
  2. General properties of modified $GH$-distances
  3. Coincidence of the semicontinuous $GH$-distance with the classical one
  4. Lower semicontinuous $GH$-distance
  5. Upper semicontinuous $GH$-distance

Key Result

Proposition 1.4

If $\sigma,\tau\in\mathcal{P}_0(X\times Y)$ and $\tau\subset\sigma$, then $\operatorname{dis}\tau\le\operatorname{dis}\sigma$.

Theorems & Definitions (49)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Theorem 1.6: BurBurIva
  • Proposition 1.7
  • Proposition 1.8
  • proof
  • Corollary 1.9
  • ...and 39 more