For a closed subset A of Euclidean space, the reach is the supremal r≥0 such that every point at distance less than r from A has a unique nearest point in A; equivalently the radius of the largest tubular neighborhood on which the nearest-point projection is well-defined and single-valued.

Principle

Quantifies extrinsic regularity by ruling out nearby self-approach and high curvature: reach controls curvature, medial axis distance and guarantees a tubular neighborhood where projection to the set is a Lipschitz (indeed smooth for smooth sets) retraction.

Demonstration

For a smooth embedded curve or hypersurface with bounded curvature, the reach is at least the reciprocal of the maximal principal curvature; for a round sphere of radius R embedded in Euclidean space the reach equals R (the distance to the medial axis being the center).

Misapplication

Using Hausdorff closeness or C^1 regularity as a substitute for positive reach — a C^1 curve can have arbitrarily small reach due to near self-approach — or confusing reach with intrinsic injectivity or convexity radii that are defined differently.

Consequence

Positive reach yields a tubular neighborhood diffeomorphic to a normal bundle, well-defined orthogonal projection, stability under perturbation scales, and enables curvature measures and geometric inference; many geometric reconstruction results require a positive lower bound on reach.

Reversal

The medial axis (set of points with multiple nearest neighbors) is the complement within the tubular neighborhood of the domain of uniqueness; reversing the notion locates where projections fail or where curvature constraints are violated.

Boundary

An extrinsic Euclidean concept applying to subsets of Euclidean space (or Riemannian manifolds via isometric embedding); it requires global control of embedding geometry and does not directly apply to abstract Riemannian manifolds without reference to an ambient Euclidean space.

Semantic Tension

Often compared to injectivity and convexity radii for embedded manifolds: reach is extrinsic and concerns nearest-point uniqueness, while injectivity and convexity radii are intrinsic geodesic notions; conflation leads to errors in reconstruction or analysis of neighborhoods.

Synthesis

The reach is the largest radius of a tubular neighborhood around an Euclidean subset on which the nearest-point map is single-valued: an extrinsic measure of boundary regularity that ties curvature bounds and medial-axis avoidance to the existence of smooth projection maps.