# The abstract concept of Duality and some related facts (part of a joint project with Shiri Artstein-Avidan)

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Vitali Milman, Tel Aviv University
Fine Hall 110

We discuss in the talk an unexpected observation that very minimal basic properties essentially uniquely define some classical transforms which traditionally are defined in a concrete and quite involved form. We start with a characterization of a very basic concept in Convexity: Duality and the Legendre transform. We show that the Legendre transform is, up to linear terms, the only involution on the class of convex lower semi-continious functions in R^n which reverses the (partial) order of functions. This leads to a different understanding of the concept of duality, which we call an abstract duality concept, and which we then apply also to many other well known settings. It is also true that any involutive transform (on this class) which exchanges summation with inf-convolution, is, up to linear terms, the Legendre transform. The classical Fourier transform may be also defined (essentially) uniquely by the condition of exchanging convolution with product together the form of the square of the transform (the last fact is a joint work also with Semyon Alesker).