Weak curvatures of irregular curves in high dimension Euclidean spaces

TL;DR

This paper introduces a robust framework for defining and analyzing weak normals and curvature measures of irregular high-dimensional curves, extending classical concepts to non-smooth cases through polygonal approximations and relaxed energies.

Contribution

It develops a new notion of weak normals for irregular curves in high dimensions, connecting discrete polygonal normals with smooth normals via relaxed total variation and energy concepts.

Findings

Weak normals are well-defined for irregular curves with finite relaxed energy.

The length of the weak normal matches the relaxed energy and satisfies an integral-geometric formula.

A curvature measure is derived from the first variation of the weak normal length.

Abstract

We deal with a robust notion of weak normals for a wide class of irregular curves defined in Euclidean spaces of high dimension. Concerning polygonal curves, the discrete normals are built up through a Gram-Schmidt procedure applied to consecutive oriented segments, and they naturally live in the projective space associated to the Gauss hyper-sphere. By using sequences of inscribed polygonals with infinitesimal modulus, a relaxed notion of total variation of the $j$-th normal to a generic curve is then introduced. For smooth curves satisfying the Jordan system, in fact, our relaxed notion agrees with the length of the smooth $j$-th normal. Correspondingly, a good notion of weak $j$-th normal of irregular curves with finite relaxed energy is introduced, and it turns out to be the strong limit of any sequence of approximating polygonals. The length of our weak normal agrees with the corresponding relaxed energy, for which a related integral-geometric formula is also obtained. We then discuss a wider class of smooth curves for which the weak normal is strictly related to the classical one, outside the inflection points. Finally, starting from the first variation of the length of the weak $j$-th normal, a natural notion of curvature measure is also analyzed.