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Topological Stability and Latschev-type Reconstruction Theorems for Spaces of Curvature Bounded Above

Rafal Komendarczyk, Sushovan Majhi, Will Tran

TL;DR

The paper addresses reconstructing the homotopy type of compact subsets with Alexandrov upper curvature by building Vietoris--Rips complexes from a Hausdorff-close Euclidean sample using a path-based metric. It introduces the large-scale distortion $\delta^{\varepsilon}_R(X)$ and shows that, under curvature bounds, one can achieve homotopy-equivalence between the sample-based complex $\mathcal{R}^{\varepsilon}_\beta(S)$ and the underlying space $X$, even when traditional sampling conditions fail due to vanishing reach or $\mu$-reach. The approach extends Latschev-type theorems to CAT$(\kappa)$ spaces, provides quantitative stability results for families of spaces, and proves reconstruction and approximation theorems that connect Hausdorff approximations to Euclidean samples with path-metric Vietoris--Rips complexes. This framework broadens the applicability of topological reconstruction to non-smooth and singular spaces, enabling finite-type reconstructions under weaker sampling assumptions and supplying stability results across Gromov--Hausdorff perturbations.

Abstract

We consider the problem of homotopy-type reconstruction of compact subsets $X\subset\R^N$ that have the Alexandrov curvature bounded above ($\leq$ $κ$) in the intrinsic length metric. The reconstructed spaces are in the form of Vietoris--Rips complexes computed from a compact sample $S$, Hausdorff--close to the unknown shape $X$. Instead of the Euclidean metric on the sample, our reconstruction technique leverages a path-based metric to compute these complexes. As naturally emerging in the framework of reconstruction, we also study the Gromov--Hausdorff topological stability and finiteness problem for general compact for subspaces of curvature bounded above. Our techniques provide novel sampling conditions as an alternative to the existing and commonly used techniques using weak feature size and $μ$--reach. To the best of our knowledge, this is the first work that establishes homotopy-type reconstruction guarantees for spaces with vanishing reach and $μ$--reach, a regime not covered by existing sampling conditions.

Topological Stability and Latschev-type Reconstruction Theorems for Spaces of Curvature Bounded Above

TL;DR

The paper addresses reconstructing the homotopy type of compact subsets with Alexandrov upper curvature by building Vietoris--Rips complexes from a Hausdorff-close Euclidean sample using a path-based metric. It introduces the large-scale distortion and shows that, under curvature bounds, one can achieve homotopy-equivalence between the sample-based complex and the underlying space , even when traditional sampling conditions fail due to vanishing reach or -reach. The approach extends Latschev-type theorems to CAT spaces, provides quantitative stability results for families of spaces, and proves reconstruction and approximation theorems that connect Hausdorff approximations to Euclidean samples with path-metric Vietoris--Rips complexes. This framework broadens the applicability of topological reconstruction to non-smooth and singular spaces, enabling finite-type reconstructions under weaker sampling assumptions and supplying stability results across Gromov--Hausdorff perturbations.

Abstract

We consider the problem of homotopy-type reconstruction of compact subsets that have the Alexandrov curvature bounded above ( ) in the intrinsic length metric. The reconstructed spaces are in the form of Vietoris--Rips complexes computed from a compact sample , Hausdorff--close to the unknown shape . Instead of the Euclidean metric on the sample, our reconstruction technique leverages a path-based metric to compute these complexes. As naturally emerging in the framework of reconstruction, we also study the Gromov--Hausdorff topological stability and finiteness problem for general compact for subspaces of curvature bounded above. Our techniques provide novel sampling conditions as an alternative to the existing and commonly used techniques using weak feature size and --reach. To the best of our knowledge, this is the first work that establishes homotopy-type reconstruction guarantees for spaces with vanishing reach and --reach, a regime not covered by existing sampling conditions.
Paper Structure (24 sections, 27 theorems, 75 equations, 2 figures)

This paper contains 24 sections, 27 theorems, 75 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background on Latschev's Theorem
  3. Related works on the reconstruction of homotopy type
  4. Acknowledgements
  5. Preliminaries
  6. Length Spaces
  7. Spaces of curvature bounded above
  8. Hausdorff and Gromov--Hausdorff Distances
  9. Abstract Simplicial Complexes and Geometric Realizations
  10. Barycentric Subdivision
  11. Latschev's Theorem For Spaces with Curvature $\leq \kappa$
  12. Hausmann's Theorem
  13. Jung's Theorem
  14. Proofs of Theorems \ref{['thm:latschev-cat']} and \ref{['thm:main-stability']}
  15. Euclidean Path Metrics and Geodesic Subspaces of RN
  16. ...and 9 more sections

Key Result

Theorem A

Let $X$ be a compact subspace of $\mathbb{R}^N$ with with curvature $\leq \kappa$ and let $\xi\in\left(0,\tfrac{1}{14}\right]$ be a fixed number. For any $\beta$ satisfying $0<\beta<\frac{1}{1+2\xi}\Delta(X)$, if we choose $0<\varepsilon\leq\beta$ such that $\delta^\varepsilon_{2\xi\beta}(X)\leq1+\l

Figures (2)

  • Figure 1: (left) Star--shaped subspace $\mathcal{S}$ ( Ninja's Star) (right) $r_{\mu}(\mathcal{S})=0$ at the cusp.
  • Figure 2: A $2$--sphere ( UFO surface) with vanishing $\mu$--reach due to the "cuspy" rim.

Theorems & Definitions (56)

  • Theorem A: Reconstruction of Subspaces with Curvature $\leq \kappa$ in $\R^N$
  • Theorem B: Quantitative Latschev's Theorem for Compact Spaces with curvature $\leq \kappa$
  • Theorem C: Topological Stability of compact Spaces with Curvature $\leq \kappa$
  • Corollary D
  • Theorem E: $(\eps,R)$--Gromov--Hausdorff closeness Theorem
  • Theorem 1.1: Quantitative Latschev's Theorem for Riemannian manifolds majhi2024demystifying
  • Theorem 1.2: Submanifold Reconstruction under Hausdorff Distance majhi2024demystifying
  • Lemma 2.1: Lower Semi-Continuity
  • Definition 2.2: Induced Length Metric
  • Definition 2.3: The space of curvature $\leq \kappa$
  • ...and 46 more