Montag, 07. Juni 2010

Kolloquiumsvortrag von Prof. Dr. Vitali Milman

Titel des Vortrags: Duality and Rigidity for Families of Convex Functions

Am Montag, dem 07. Juni 2010 hält 

Prof. Dr. Vitali Milman (University of Tel Aviv)

um 16:45 Uhr im Hörsaal D2 einen Vortrag mit dem Thema

"Duality and Rigidity for Families of Convex Functions"

Zu dieser Veranstaltung sind alle Interessenten herzlich eingeladen.

Um 16:15 Uhr trifft man sich zur Begrüßung des Gastes bei Tee und Kaffee im Besprechungsraum D2.343.

Abstract:

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 and Functional Analysis: 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, 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.
In the same time, considering the class of non-negative convex functions(with 0 value at 0), changed the picture and brings an additional, second duality for this class, which was not considered before. We will study this new duality and show that there are exactly two dualities on this class. In particular, this leads to new structures on this class.
We analyse the first step of the proof of the above results and we quickly turn it to the study of rigidity of the identity map on the family of convex functions.
The classical Fourier transform may be also defined (essentially) uniquely by the condition of exchanging convolution with product together with the form of the square of the transform (the last fact is a joint work also with Semyon Alesker). The talk will be understandable for graduate students.
(Joint work with  Shiri Artstein-Avidan).

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