Lower bounds for the reach and applications

TL;DR

This paper presents a practical method to compute guaranteed lower bounds for the reach of submanifolds defined by smooth functions, with applications in topology, geometry, and spectral theory.

Contribution

It introduces a rigorous, numerically verified algorithm for lower bounding the reach of smooth submanifolds, applicable to high-dimensional and non-polynomial cases.

Findings

Developed an algorithm for guaranteed reach bounds of submanifolds

Applied the method to compute homology of planar curves

Provided bounds on eigenvalues and deformation stability of varieties

Abstract

The reach of a submanifold of $\mathbb{R}^N$ is defined as the largest radius of a tubular neighbourhood around the submanifold that avoids self-intersections. While essential in geometric and topological applications, computing the reach explicitly is notoriously difficult. In this paper, we introduce a rigorous and practical method to compute a guaranteed lower bound for the reach of a submanifold described as the common zero-set of finitely many smooth functions, not necessarily polynomials. Our algorithm uses techniques from numerically verified proofs and is particularly suitable for high-performance parallel implementations. We illustrate the utility of this method through several applications. Of special note is a novel algorithm for computing the homology groups of planar curves, achieved by constructing a cubical complex that deformation retracts onto the curve--an approach potentially extendable to higher-dimensional manifolds. Additional applications include an improved comparison inequality between intrinsic and extrinsic distances for submanifolds of $\mathbb{R}^N$, lower bounds for the first eigenvalue of the Laplacian on algebraic varieties and explicit bounds on how much smooth varieties can be deformed without changing their diffeomorphism type.