An algorithm for determining minimax approximations to strictly convex functions by means of first degree splines with free knots is proposed. The method may be used in one of two modes. Either the minimax approximation having a prescribed number of knots can be found, or the minimax approximation with the smallest number of knots whose maximum error does not exceed a given error can be determined. The method, while processing quadratic convergence, does not require a knowledge of derivatives higher than the first.