Abstract. The paper proves the minimax theorem for a specific class of functions that are defined on branching polylines in a linear space, not on convex subsets of a linear space. The existence of a saddle point for such functions does not follow directly from the classical minimax theorem and needs individual consideration based both on convex analysis and on graph theory. The paper presents a self-sufficient analysis of the problem. It contains everything that enables plain understanding of the main result and its proof and avoids using concepts outside the scope of obligatory mathematical education of engineers. The paper is adressed to researchers in applied mechanics, engineering and other applied sciences as well as to mathematicians who lecture convex analysis and optimization methods to non-mathematicians.

Keywords: minimax, saddle point, convex analysis, optimization, branching polyline.