Interpolation of medians (Q1805833)

Given \(n+1\) disjoint bounded intervals \(I_0, \ldots ,I_n\) and \(n+1\) real numbers \(\alpha_0,\ldots ,\alpha_n\), a polynomial is to be determined such that its median in the interval \(I_j\) equals \(\alpha_j\) \((j=0,\ldots ,n)\). Although there is a connection with \(L_1\)-approximation, this is a nonlinear ``interpolation problem''. Topological methods are used for proving the existence of a unique solution. The method is not restricted to polynomials and applies to more families of functions.