We describe a numerical shape optimization method of continua that minimizes maximum local measure such as stress and displacement. A solution to this min-max problem subject to volume constraint is proposed. To avoid impossibility of differentiation, local functionals are transposed to global integral functionals using the Kreisselmeier-Steinhauser function. With this function, a multiple leading problem is also transposed to a single loading problem. The shape gradient functions which are applied to the traction method are theoretically derived using the Lagrange multipliers and the material derivative method. Using the traction method, the optimum domain variation to decrease the objective functional is numerically and iteratively determined that maintaining the smoothness of the boundaries. The calculated results of 2D and 3D examples show the effectiveness and practical utility of the proposed method for min-max problems in shape designs.