Interaction of order and convexity

Abstract. This is a talk on interaction of the techniques of positivity and abstract convexity in functional analysis and the extremal problems of convex geometry. Alexandr Danilovich Alexandrov became the first and foremost Russian geome-ter of the twentieth century. He contributed to mathematics under the slogan: “Retreat to Euclid, ” remarking that “the pathos of contemporary mathematics is the return to Ancient Greece. ” Hermann Minkowski revolutionized the theory of numbers with the aid of the synthetic geometry of convex surfaces. The ideas and techniques of the geometry of numbers comprised the fundamentals of functional analysis which was created by Banach. The pioneering studies of Alexandrov con-tinued the efforts of Minkowski and enriched geometry with the methods of measure theory and functional analysis. Alexandrov accomplished the turnround to the an-cient synthetic geometry in a much deeper and subtler sense than it is generally acknowledged today. Geometry in the large reduces in no way to overcoming the lo-cal restrictions of differential geometry which bases upon the infinitesimal method