Analytic and algebraic related structures on the families of convex sets and log-concave functions

Friday, November 6, 2015 - 4:00pm
217 Strickland
Vitali Milman

The main goal of the talk is to show how some classical constructions in
Geometry and Analysis appear (and in a unique way) from elementary and  
very simplest properties. For example, the polarity relation and support functions are very important
and well known constructions in Convex Geometry, but what elementary  
property uniquely implies this construction(s), and what would be  
their functional versions, say, in the class of log-concave functions?  
And yes, they are uniquely defined also for this class, as well as for  
many other classes of functions.
  Another example: How can one identify the "square root of a convex body"? Yes, it is
possible (however, "the square of a convex body" does not exist in general).

  In this talk we will mostly deal with Geometric results of this
nature. We also construct summation operation on the class of  
log-concave functions which polarizes the Lebesgue Integral and  
introduces the notion of a mixed integral parallell to mixed volumes  
for convex bodies. We will show some inequalities coming from Convex  
geometry, but which are presented already on the level of log-concave  
(or even more generally quasi-concave) functions.
  In the end we will characterize the Fourier transform (on the  
Schwartz class in R^n) as, essentially, the only map which transforms  
the product to the convolution. There is an exciting continuation of  
this direction in Analysis but we will stop on this example.

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