Subdifferentials of Convex Functions
The crucial discovery of the concept of differential calculus is due to Pierre de Fermat (1601–1655), who was one of the most important innovators in the history of mathematics. It is to him that we owe a rule for determining extrema, described, without proof, in a short treatise Methodus ad disquirendam Maximum et Minimam written in 1637. The importance of his discoveries in number theory has eclipsed the contributions which this exceptional and modest man made to other areas of mathematics. Fermat also was the first to discover the “principle of least time” in optics, the prototype of the variational principles governing so many physical and mechanical laws. He shared independently with Descartes the invention of analytic geometry and with Pascal the creation of the mathematical theory of probability. His achievements in number theory overshadowed his other contributions, as the Last Fermat Theorem which remained a challenge for such a long time, and still is a challenge if indeed the simple proof of Piere de Fermat did exist. Not to mention his compositions in French, Latin, Italian and Spanish verse and his Grecian erudition. It is also notable that he was able to find time for these occupations in the midst of his duties as counsellor to the parliament of Toulouse (even taking into account Fermat’s genius, this makes us reflect on the leisure activities offered by a lawyer’s career).
KeywordsConvex Function Normal Cone Tangent Cone Differential Calculus Legendre Transformation
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