On Radon transforms between lines and hyperplanes

We obtain new inversion formulas for the Radon transform and its dual between lines and hyperplanes in [Formula: see text]. The Radon transform in this setting is non-injective and the consideration is restricted to the so-called quasi-radial functions that are constant on symmetric clusters of lines. For the corresponding dual transform, which is injective, explicit inversion formulas are obtained both in the symmetric case and in full generality. The main tools are the Funk transform on the sphere, the Radon-John [Formula: see text]-plane transform in [Formula: see text], the Grassmannian modification of the Kelvin transform, and the Erdélyi–Kober fractional integrals.

Lp−Winterniz problem on firey projection of convex bodies

For $p\geq 1$, Lutwak, Yang and Zhang introduced the concept of $p$-projection body, and Lutwak introduced the concept of $L_{p}-$ affine surface area of convex body. In this paper, we develop the Minkowski-Funk transform approach in the $L_{p}$-Brunn-Minkowski theory. We consider the question of whether $\Pi_{p}K\subseteq \Pi_{p}L$ implies $\Omega_{p}(K) \leq \Omega_{p}(L)$, where $\Pi_{p}K$ and $\Omega_{p}K$ denotes the $p-$projection body of convex body $K$ and the $L_{p}-$affine surface area of convex body $K$, respectively. We also formulate and solve a generalized $L_{p}-$Winterniz problem for Firey projections.

The Funk transform as a Penrose transform

The Funk transform is the integral transform from the space of smooth even functions on the unit sphere S2⊂ℝ3 to itself defined by integration over great circles. One can regard this transform as a limit in a certain sense of the Penrose transform from [Copf ]ℙ2 to [Copf ]ℙ*ast;2. We exploit this viewpoint by developing a new proof of the bijectivity of the Funk transform which proceeds by considering the cohomology of a certain involutive (or formally integrable) structure on an intermediate space. This is the simplest example of what we hope will prove to be a general method of obtaining results in real integral geometry by means of complex holomorphic methods derived from the Penrose transform.