Title:
Extremizers for the Radon Transform: A Symmetric Story
<br><br>
Abstract:
The Radon transform R takes a function whose domain is Euclidean space
R^d for some dimension d greater than one, and forms its integrals over
all hyperplanes. R and its inverse are widely used in medical imaging.
<br><br>
R satisfies certain inequalities in terms of L^p norms, with exponents
dictated by the dimension d.
This talk will discuss the nature of those functions which extremize
the most fundamental
of these inequalities.  It turns out that the Radon transform enjoys a
remarkably large group of symmetries, which can be used to determine
the extremizers in closed form.  The proof, in which symmetries arise
in at least three distinct ways, will be outlined.
<br><br>
Inequalities in L^p norms are among the most basic gadgets in any
analyst's toolchest.
Proofs often involve decomposition, synthesis, application of the
triangle inequality,and combinations of simpler inequalities which are optimal
individually, but not in combination. Such proofs rarely yield optimal
constants or reveal the identity
of extremizing functions. The talk will briefly review some classical
inequalities from this perspective.