A characterization of Chebyshev curves in \(R^ n\) (Q1900097)

Given a totally ordered set \(T\) containing \(n+1\) elements \(t_i\), \(i = 0,\dots,n\), the curve \(a: T\to\mathbb R^n\) is called Chebyshev curve if the determinant \(\det(a(t_0),\ldots,a(t_n))\) is either positive or negative for \(t_0 < \cdots < t_n\). For finite \(T\) characterisations of these curves are well-known. The author gives here a characterisation for non-finite sets \(T\), which is gained by extending the known results. This is done by observing the changes of signs of the scalar product of the vectors \(a(t)\) and arbitrary vectors \(x\) in \(\mathbb R^n\).