The Inequalities

Comparing quantities is one of the most natural activities: who has the fewest objects, or the largest field? Before knowing all the mathematical tricks that allow one to increase the area of one's land or a precious capital, it is useful to return to the geometric origins of the question, in order to properly visualize the quantities at play. The triangle inequality and Dido's trick still amaze us today through their fruitfulness.

Improve, gain a little here, avoid losing too much there, optimize… all of this falls under the purview of analysis! If geometric intuition helps us compare, it is mathematical analysis that will allow us to quantify variations and thus to bound from above or below many expressions. Among the stars of the field, we find the arithmetic-geometric mean inequality, well known to high school students, and the Cauchy–Schwarz inequality, more advanced but of formidable power.

In many mathematical fields, inequalities play an important role. This is the case for example in probability, with the Bienaymé‒Tchebychev inequality, or in all that relates to optimization. Many physical phenomena are also modeled by inequalities; thermodynamics gives us a striking example. In economic and social sciences, the study of inequalities is a central topic, whether they concern wealth, labor, housing, education…