Severi-Bouligand tangents, Frenet frames and Riesz spaces

Abstract

It was recently proved that a compact set $X\subseteq {\mathbb{R}}^2$ has an outgoing Severi-Bouligand tangent vector $u\ne 0$ at $x\in X$ iff some principal ideal of the Riesz space $\mathcal{R}(X)$ of piecewise linear functions on $X$ is not an intersection of maximal ideals. "Outgoing" means $X\cap [x,x+u]=\{x\}$. Suppose now $X\subseteq {\mathbb{R}}^n$ and some principal ideal of $\mathcal{R}(X)$ is not an intersection of maximal ideals. We prove that this is equivalent to saying that $X$ contains a sequence $\{x_i\}$ whose Frenet $k$-frame $(u_1,\dots ,u_k)$ is an outgoing Severi-Bouligand tangent of $X$. When the $\{x_i\}$ are taken as sample points of a smooth curve $\gamma ,$ the Frenet $k$-frames of $\{x_i\}$ and of $\gamma$ coincide. The computation of Frenet frames via sample sequences does not require the knowledge of any higher-order derivative of $\gamma$.

References in corpus (1)

• Unital hyperarchimedean vector lattices