The Radon transform forms the integral of a function over all affine hyperplanes in Euclidean space R^d. It satisfies various L^p to L^q inequalities in Lebesgue space norms. One of these inequalities has connections with several other topics, including a certain convolution operator, the Kakeya problem, and a multilinear inequality involving determinants. It enjoys an exceptionally large group of symmetries.
We discuss inverse questions about functions which exactly or nearly extremize this inequality. In particular, (all) extremizers have recently been identified. In this longish story, a leading role is played by considerations of symmetry. A remarkable happenstance is the existence of equivalent formulations for three different operators; each incarnation reveals its own facet of the full symmetry group.
We will outline the steps which lead to this identification. Along the way we will touch briefly on combinatorics, equicontinuity, weighted norm inequalities, a nonlinear and nonlocal Euler-Lagrange equation, rearrangement inequalities, the Brunn-Minkowski inequality in one dimension, and the Hardy-Littlewood-Sobolev inequality. All would still be for nought, were it not for the timely appearance of one more (conformal) symmetry.
