Finite Good Witnesses for Generalized Curve Projections at the Rectifiable Endpoint

Abstract

For a $1$-rectifiable set $E \subset \mathbb{R}^d$ of positive length, a theorem of Federer shows that, among any $d$ linearly independent orthogonal projections of $E$, at least one has positive length. We develop a version of this finite-witness principle for generalized curve projections. Given scalar-valued mappings $\varphi_1,\ldots,\varphi_d:\mathbb{R}^d\longrightarrow\mathbb{R}$, we introduce the canonical encoding map $\mathsf{H}:=(\varphi_1,\ldots,\varphi_d)$. Where $D\mathsf{H}$ is invertible, a local bilipschitz change of variables and Federer's theorem show that $\varphi_j(E)$ has positive length for some $j$. If a positive-length portion of $E$ lies in the critical set, we instead argue intrinsically on a $C^1$ hypersurface containing it, provided that $\mathsf{H}$ retains full tangential rank there. This yields an abundance of deterministic finite witnesses, as well as structural bounds for those exceptional parameters where the good witness property fails. A new fold non-degeneracy condition makes our constructions stable under perturbations of the underlying parameters. At this rectifiable endpoint, these conclusions complement work of Peres--Schlag, which developed exceptional set estimates for fractal sets of Hausdorff dimension strictly greater than one. We then apply our framework to several nonlinear projection families. Affinely independent pinned squared-distance maps satisfy the tangential and fold conditions, whereas two planar radial projections lose tangential rank along their critical line. We also verify the hypotheses for more exotic examples, including nonlinear anisotropic distances, Bregman functionals arising from smooth approximations of polyhedral norms, and families whose critical hypersurfaces have prescribed $C^2$ geometry.