# Extremizers of a Radon Transform Inequality

# Extremizers of a Radon Transform Inequality

This chapter discusses the extremizers of a Radon transform inequality. The model for this analysis is Lieb's characterization of extremizers for the Hardy–Littlewood–Sobolev inequality for certain pairs of exponents. The chapter first introduces the four main steps of this model and sets up an endpoint inequality, before developing the identities to be used for the analysis in the remainder of this chapter. It then discusses some preliminary facts concerning extremizers and brings up direct and inverse Steiner symmetrization. Finally, the chapter returns to the inequality described in the first part of the chapter and begins the process of identifying extremizers for it. It concludes with further discussion on compact subgroups of the affine group as well as critical points.

*Keywords:*
extremizers, inequalities, Radon transform, Radon transform inequality, Steiner symmetrization, affine group, critical points

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