Convexity
Implicitly present in the geometry of ancient Greece, convexity finds its rigorous formulation in the 19th century in analysis, within the framework of solving inequalities. In all the fields that use it, as soon as the object being manipulated is convex, we observe that everything becomes easy! Subtle geometric properties "leap to the eye" or are easily established. The concept invades many mathematical fields (topology, operations research...) before being used well beyond, as in artificial intelligence, finance or economics.
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A modern theory with ancient roots | Tangente
It was during the twentieth century that convexity emerged as a mathematical discipline in its own right. Before then, eminent mathematicians had occasionally glimpsed the potential value of this notion in geometry and analysis.
Optimizing consumption | Tangente
Making consumption choices is no easy matter. But for a rational consumer, the notion of convexity is extremely useful! Consumer theory provides a case in point here.
Convex geometry
Convex geometry lies at the crossroads of optimization, analysis, topology, combinatorics and, of course, geometry. The graphical and visual interpretations it affords are powerful aids to intuition. Yet fundamental questions remain open.
Useful functions in analysis
With a graph that "looks skyward," convex functions give us reason for optimism. More seriously, first introduced to prove elegant inequalities, they have shown their importance far beyond mathematics. But what, exactly, are the properties of convex functions?
Convexity and inequalities
Convex functions arose in analysis as a means of proving inequalities
Convexity in finance | Tangente
Mathematics plays an important role in finance. In recent years, returns on capital have been drastically reduced for various economic and political reasons. In Europe today, negligible returns on capital are becoming the norm, yet inflation remains very real, at around 5%.
Beyond convexity | Tangente
The intuitive idea behind convexity seems entirely natural. Yet one cannot help wondering whether slightly altering the formal definition might lead to other interesting ideas… Here too, convexity is just as fruitful!