  In this paper we demonstrate that a well known linear inequality method
developed for rational Chebyshev approximation is equivalent to the application
of the bisection method used in quasiconvex optimisation. Although this
correspondence is not surprising, it naturally connects rational and
generalised rational Chebyshev approximation problems with modern developments
in the area of quasiconvex functions and therefore offers more theoretical and
computational tools for solving this problem. %In particular, this observation
leads a straightforward extension of the linear inequality method to
generalised rational approximation in the sense of Loeb (ratio of two linear
forms). The second important contribution of this paper is the extension of the
linear inequality method to a broader class of Chebyshev approximation
problems, where the corresponding objective functions remain quasiconvex. In
this broader class of functions, the inequalities are no longer required to be
linear: it is enough for each inequality to define a convex set and the
computational challenge is in solving the corresponding convex feasibility
problems. Therefore, we propose a more systematic and general approach for
treating Chebyshev approximation problems. In particular, we are looking at the
problems where the approximations are quasilinear functions with respect to
their parameters, that are also the decision variables in the corresponding
optimisation problems.
