A short proof of the snake theorem of Karlin (Q812545)

Karlin's `snake theorem' [see \textit{S. Karlin} and \textit{W. J. Studden}, Tchebycheff Systems with Applications in Analysis and Statistics, Pure and Applied Mathematics. Vol. 15. New York etc.: Intersecience Publishers a division of John Wiley \& Sons (1966; Zbl 0153.38902), Chapter~2] is one of the most important results of the theory of Tchebycheff systems (T-systems). Karlin's proof of this theorem is based on the Brouwer fixed-point theorem and the possibility of approximating T-systems by differentiable T-systems. Using a new result on surjective mappings on simplices, the present paper obtains a considerably simplified proof of the snake theorem which avoids the approximation techniques of Karlin's proof. The new proof also establishes the existence of a snake for more than two boundary curves.