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.