Lower bounds for the reach and applications

TL;DR

The paper tackles the challenge of obtaining a guaranteed lower bound for the reach $\tau$ of a smooth submanifold $M\subset\mathbb{R}^N$ defined as a zero set, by developing a subdivision-based, numerically verified framework that first bounds the $L^1$-norm of the gradient on $M$ and then bounds curvature and bottlenecks to yield a computable $\tau$. The main approach integrates a rigorous gradient-bound algorithm with geometric estimates, and extends naturally from single to multiple defining equations $f_i$, via the Gram matrix $g$ and its determinant. The authors demonstrate multiple applications: improved extrinsic-intrinsic distance estimates, homology computation for planar curves via deformation retracts of selected boxes, lower bounds for Laplacian eigenvalues, and effective bounds for deforming algebraic varieties without changing their diffeomorphism type. The work enables reliable geometric and topological analysis from smooth (not necessarily polynomial) defining data and is amenable to parallel high-performance computation, broadening practical reach in computational geometry and topology.

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.

Figures (4)

• Figure 1: \ref{['algorithm:lower-bound-for-df']} being executed for the function $f(x,y)=x^2+y^2-1$ on $[-M_1,M_1]$ for $M_1=2$. Boxes in NewBoxes are drawn with a gray border, boxes in CaseOneBoxes are drawn with a green border, and boxes in CaseTwoBoxes are drawn with a red border. The algorithm takes $213$ steps to terminate, and we show progress at 12 steps throughout the process.
• Figure 2: Picture of a variety $M=Z(f) \subset \mathbb{R}^2$ in red. Red shaded boxes are selected according to \ref{['definition:selected-box']}. The blue shaded box contains a point of $M$ but has not been selected. The conclusion of \ref{['corollary:boxes-and-M-have-same-homology-groups']} remains true: the union of all selected boxes has the same Betti numbers as $M$.
• Figure 3: A constellation of points attaining the maximum value for $\angle(q-y, p-y)$, namely $\angle(q-y, p-y) = 2 \arctan (1/2)$.
• Figure 4: Assume a box which contains at least a point of $M$ and it is not selected, has two faces with a non-empty intersection with $M$. There are two possible cases, which are treated separately in the proof of \ref{['proposition:not-selected-box-has-at-most-one-nonempty-face']}. Case 1, in which $M$ meets opposite faces of the box, is displayed on the left. Case 2, in which $M$ meets adjacent sides of the box, is displayed on the right.

Theorems & Definitions (71)

• Proposition 1.1
• Corollary 1.2
• Definition 2.1
• Definition 2.2: Federer1959
• Proposition 2.4
• proof
• Definition 2.5
• Definition 2.7
• Proposition 2.8
• Definition 2.9