From measuring tool to geometrical object: Minkowski's development of the concept of convex bodies (Q2468669)

T. H. Kjeldsen gives a detailed analysis of an important chapter in the history of the modern theory of convex sets. In that history the German mathematician Karl Hermann Brunn (1862--1939) is usually mentioned as the first who was engaged in systematic studies about convex sets, followed by Hermann Minkowski (1864--1909). Kjeldsen points out that Minkowski did not know about Brunn's work until after he had begun his own investigations regarding convex sets. The author states further that the modern theory of convex sets grew out of Minkowski's work. Therefore the main aim of the paper is to explain: ``why and how the concept of convex bodies emerged, took form, and led to the beginning of a theory of convexity in Minkowski's mathematical practice.'' As a result of her analysis Kjeldsen characterizes three phases in Minkowski's mathematical practice leading to a theory of convex bodies. The first one she describes by geometrical treatment of the minimum problem for positive definite quadratic forms. She argues further that Minkowski was led in the second phase by elaborating his geometrical approach into a more general method to the introduction of nowhere concave bodies with middle point (``Eichkörper'') and a distance function as measuring tools. Finally, the third phase was characterised by Minkowski's investigations of convex bodies as geometrical objects for its own sake. Summarizing this development of new knowledge the author sketches the process as an interplay between two research strategies of (1) answering an old question with new methods and (2) posing and answering new questions for new mathematical objects.