Morse theory of Euclidean distance functions from algebraic hypersurfaces

TL;DR

The paper develops a Morse-theoretic framework for the Euclidean distance function $dist_Y|_X$ when $X$ and $Y$ are (generically) smooth algebraic hypersurfaces, addressing the non-differentiability of distance by employing Clarke subdifferentials and continuous selections. It introduces a two-index nondegenerate theory with a piecewise linear index $k(x)$ and a quadratic index $\iota(x)$, yielding topological control of sublevel sets and a deformation mechanism across critical values; it also proves finite and generically nondegenerate critical points via a multijet transversality approach. The work unifies bottleneck and Euclidean distance degree notions, providing bounds on the number of critical points and enabling a relative Euler-characteristic formula that links algebraic geometry with Lipschitz-Morse theory. The results give practical tools for computational algebraic geometry tasks such as bottleneck analysis and distance optimization on hypersurfaces, with robust genericity guarantees and explicit complexity bounds. Overall, the framework advances a rigorous, definable setting for topological analysis of distance-based problems on algebraic varieties and opens pathways for algorithmic applications in geometry-aware data analysis and tensor approximation.

Abstract

Let $Y\subseteq \mathbb{R}^n$ be a closed definable subset and $X\subseteq \mathbb{R}^n$ be a smooth manifold. We construct a version of Morse Theory for the restriction to $X$ of the Euclidean distance function from $Y$. This is done using the notion of critical points of Lipschitz functions and applying the theory of continuous selections. In this theory, nondegenerate critical points have two indices: a quadratic index (as in classical Morse Theory), and a piecewise linear index (that relates to the notion of bottlenecks). This framework is flexible enough to simultaneously treat and unify the study of two cases of interest for computational algebraic geometry: bottlenecks and nearest point problems. We provide a technical toolset guaranteeing the applicability of the theory to the case where $X, Y$ are generic algebraic hypersurfaces and use it to bound the number of critical points of the distance from $Y$ restricted to $X$, among other applications.

Figures (4)

• Figure 1: Example of the subdifferential of $f = {\mathrm{dist}_Y|_X}$ at a critical point $x$. The subdifferential $\partial_x f$ is the interval between the projections of the vectors $\frac{x - y_0}{\| x - y_0 \|}$ and $\frac{x - y_1}{\| x - y_1 \|}$ onto $T_x X$.
• Figure 2: Example in $\mathbb{R}^3$ of the characterization of critical points using normal spaces. $X$ is a sphere and $Y = \{y_0, y_1\}$ consists of two points. The plane represents the medial axis $M_Y$; the small arrows represent normal vectors from various points of $X$; $N_{x_1} X$ (the horizontal dashed line) is the normal space of $X$ at $x_1$.
• Figure 3: Examples of critical points of $f = {\mathrm{dist}_Y|_X}$ and their degeneracy. (\ref{['F:sub-nondegen']}) and (\ref{['F:sub-degen1']}): $Y$ is a finite set of coplanar points in $\mathbb{R}^3$ and $X$ is a line that intersects the plane of the paper at a single point $x$. (\ref{['F:sub-degen2']}): $X$ and $Y$ are lines.
• Figure 4: Different examples of ${\mathrm{dist}_Y|_X}$ with different piecewise linear and quadratic indices, where $Y$ is a finite set of points or a circle and $X$ is a line. Indices are written as $(k(x), \iota(x))$.

Theorems & Definitions (92)

• Example 1.1.1
• Definition 1.2.1: Critical points, \ref{['D:critical']}
• Proposition 1.2.2: Deformation Lemma, \ref{['lem:deformation']}
• Definition 1.3.1: Nondegenerate critical points, \ref{['D:nondegenerate-crit-pt']}
• Proposition 1.3.2: \ref{['propo:criticalpass2dist']}
• Theorem 1.4.1: \ref{['T:Yomdin']}
• Definition 1.4.2: $k$-critical points, \ref{['D:k-crit-pt']}
• Theorem 1.4.3
• Remark 1.4.4
• Remark 1.4.5: A duality formula