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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