Summary: An examination is made of a linear antagonistic differential game between two persons with a fixed instant of termination, with a convex payoff function, and with geometrical constraints on the controls of the players. The control of the first player, who minimizes the values of the payoff, is assumed to be scalar. A method is indicated for constructing a universal positional strategy of the first player, that, in the absence of approximation errors, is an optimal universal strategy. An estimate of the guarantee of the first player when he uses the described strategy in a discrete control scheme is included. The possibilities of numerical constructions are demonstrated with two examples, where the strategy of the first player is realized with the aid of a computer.