Multidimensional Geometry, Functions of Very Many Variables, and Probability

The diverse facts (from statistical physics, probability theory, dynamical systems, geometry, analysis) which feature in this paper are known to the experts, but as independent, isolated observations. What we do here is interpret and discuss them as instants of the same phenomenon of measure concentration, which occurs in multidimensional spaces and holds for functions of very many variables. Such functions show a tendency towards being almost constant from the point of view of the observer. We also prove a general theorem on the concentration of the volume in a multidimensional domain.