Severi–Bouligand tangents, Frenet frames and Riesz spaces

A compact set X⊆R2X⊆R2 has an outgoing Severi–Bouligand tangent unit vector u   at some point x∈Xx∈X iff some principal quotient of the Riesz space R(X)R(X) of piecewise linear functions on X is not archimedean. To generalize this preliminary result, we extend the classical definition of Frenet k  -frame to any sequence {xi}{xi} of points in RnRn converging to a point x  , in such a way that when the {xi}{xi} arise as sample points of a smooth curve γ, the Frenet k  -frames of {xi}{xi} and of γ at x coincide. Our method of computation of Frenet frames via sample sequences of γ does not require the knowledge of any higher-order derivative of γ  . Given a compact set X⊆RnX⊆Rn and a point x∈Rnx∈Rn, a Frenet k-frame u is said to be a tangent of X at x if X   contains a sequence {xi}{xi} converging to x, whose Frenet k-frame is u. We prove that X has an outgoing k-dimensional tangent of X   iff some principal quotient of R(X)R(X) is not archimedean. If, in addition, X is convex, then X has no outgoing tangents iff it is a polyhedron.