From measuring tool to geometrical object: Minkowski’s development of the concept of convex bodies

Abstract

The present paper is concerned with the emergence of the modern theory of convex sets. Whereas the special instances of what we today understand by convex sets, such as the circle or regular polygons, have been studied throughout the history of mathematics, the modern theory understood as the systematic study of sets characterised exclusively by the property of convexity began only towards the end of the nineteenth century from where it developed into one of the many new disciplines of twentieth century mathematics. Today the theory of convexity is considered a central theory not least due to its expansion into almost all important areas of mathematics such as geometry, analysis, and applied mathematics.1 According to the history2 presented in textbooks on the theory of convexity the German mathematician Karl Hermann Brunn (1862–1939) was the first to engage in such systematic studies. His studies were then followed by the work of another German mathematician, Hermann Minkowski (1864–1909), who developed the theory further and explored some of its many applications. Examining this history one realises that Minkowski did not know about Brunn’s work until after he himself had begun his own investigations of what led to the theory of convexity. And even though Bonnesen