Hugo Steinhaus - Mathematical Contributions

Mathematical Contributions

Steinhaus authored over 170 works. Unlike his student, Stefan Banach, who tended to specialize narrowly in the field of functional analysis, Steinhaus made contributions to a wide range of mathematical sub-disciplines, including geometry, probability theory, functional analysis, theory of trigonometric and Fourier series as well as mathematical logic. He also wrote in the area of applied mathematics and enthusiastically collaborated with engineers, geologists, economists, physicians, biologists and, in Kac's words, "even lawyers".

Probably his most notable contribution to functional analysis was the 1927 proof of the Banach–Steinhaus theorem, given along with Stefan Banach, which is now one of the fundamental tools in this branch of mathematics.

His interest in games led him to propose an early formal definition of a strategy, anticipating John von Neumann's more complete treatment of few years later. As a result he is regarded as one of the early founders of modern game theory. As a result of his work on infinite games Steinhaus, together with another of his students', Jan Mycielski, proposed the Axiom of determinacy.

Steinhaus was also among the early contributors and founders of the probability theory, which at the time was still in its infancy and was not even considered to be an actual part of mathematics. He provided the first axiomatic measure-theoretic description of coin-tossing, which was to influence the full of axiomatization of probability by the Russian mathematician Andrey Kolmogorov a decade later. He was also the first to offer precise definitions of what it means for two events to be "independent", as well as for what it means for a random variable to be "uniformly distributed".

While in hiding during World War II Steinhaus worked on the so-called "cake cutting problem"; how to divide a given resource in a manner which is "fair" according to precisely defined criteria, such as proportionality and envy-free.

Steinhaus was also the first person to conjecture the Ham sandwich theorem, and one of the first to propose the method of k-means clustering