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    Severi-Bouligand tangents, Frenet frames and Riesz spaces

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    It was recently proved that a compact set X⊆R2X\subseteq \mathbb R^2 has an outgoing Severi-Bouligand tangent vector u≠0u\not=0 at x∈Xx\in X iff some principal ideal of the Riesz space R(X)\mathcal R(X) of piecewise linear functions on XX is not an intersection of maximal ideals. "Outgoing" means X∩[x,x+u]={x}X\cap [x,x+u]=\{x\}. Suppose now X⊆RnX\subseteq \mathbb{R}^n and some principal ideal of R(X)\mathcal R(X) is not an intersection of maximal ideals. We prove that this is equivalent to saying that XX contains a sequence {xi}\{x_i\} whose Frenet kk-frame (u1,…,uk)(u_1,\ldots,u_k) is an outgoing Severi-Bouligand tangent of XX. When the {xi}\{x_i\} are taken as sample points of a smooth curve γ,\gamma, the Frenet kk-frames of {xi}\{x_i\} and of γ\gamma coincide. The computation of Frenet frames via sample sequences does not require the knowledge of any higher-order derivative of γ\gamma
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