[Paper Review] On the Minkowski-Funk Transform

Susanna Dann|arXiv (Cornell University)|Mar 26, 2010

Optical measurement and interference techniques2 references3 citations

TL;DR

This paper presents a comprehensive study of the Minkowski-Funk transform, establishing an inversion formula for the transform and connecting it to classical problems in integral geometry and convex geometry. It proves that a function on the sphere is constant if and only if its Minkowski-Funk transform is constant, thereby offering a new analytical proof of Minkowski's theorem on surface area measures using spherical harmonics and fractional integral operators.

ABSTRACT

The subject of this paper is the history of the Minkowski-Funk Transform. After introducing the Minkowski-Funk Transform as well as its dual transform and a generalization of both, we will present an inversion formula of the Minkowski-Funk Transform. Then we will discuss the history of this problem: related work by Minkowski and Funk and the connection between their work.

Motivation & Objective

• To establish a rigorous inversion formula for the Minkowski-Funk transform on the sphere $ S^{n-1} $.
• To clarify the historical connection between Minkowski’s work on surface area measures and Funk’s integral transforms.
• To generalize the Minkowski-Funk transform using rotation-invariant measures and fractional integral operators.
• To provide a new analytical proof of Minkowski’s theorem on the existence of convex bodies with prescribed surface area measures.
• To demonstrate that the Minkowski-Funk transform preserves symmetry and commutes with rotations, enabling harmonic analysis techniques.

Proposed method

• The Minkowski-Funk transform is defined as $ (Mf)(\xi) = \int_{\xi} f(x)\,dx $, where $ \xi \subset S^{n-1} $ is an $ (n-2) $-dimensional geodesic, with a rotation-invariant measure.
• The dual transform $ M^*\varphi $ is defined by integrating $ \varphi $ over all geodesics passing through a point $ x \in S^{n-1} $.
• A generalized transform $ M_\theta f $ is introduced, integrating $ f $ over all points at geodesic distance $ \theta $ from a given geodesic $ \xi $.
• The inversion formula is derived using Riemann-Liouville fractional integrals and Catalan’s formula for spherical integrals.
• Spherical harmonics decomposition is applied to show that $ Mf \equiv \text{constant} $ if and only if $ f \equiv \text{constant} $.
• The proof leverages the fact that the kernel of the Minkowski-Funk transform consists of odd functions, and uses the identity $ \widetilde{M}^{\alpha} \widetilde{M}^{\beta} f = f $ when $ \alpha + \beta = 2 - n $.

Experimental results

Research questions

• RQ1 How can the Minkowski-Funk transform be inverted, and what is the role of fractional integrals in this inversion?
• RQ2 What is the precise connection between Minkowski’s surface area measure problem and the Funk transform?
• RQ3 Under what conditions does the Minkowski-Funk transform of a function $ f $ being constant imply that $ f $ itself is constant?
• RQ4 How do rotation-invariant measures and spherical harmonics contribute to the analysis of the Minkowski-Funk transform?
• RQ5 Can the Minkowski-Funk transform be generalized to include distance-level sets, and what are the implications for inversion?

Key findings

• The inversion formula for the Minkowski-Funk transform is given by $ (M_\theta^* Mf)(x) = 2\pi^{\frac{n-2}{2}} \cos^{3-n}\theta \cdot (I_{0+}^{\frac{n-2}{2}} \widetilde{f}_x)(\cos^2\theta) $, where $ \widetilde{f}_x(\tau) = \frac{1}{\sqrt{\tau}} (M^{\sqrt{\tau}} f)(x) $.
• The Minkowski-Funk transform commutes with rotations, which allows the use of group-theoretic and harmonic analysis techniques on $ SO(n) $.
• The kernel of the Minkowski-Funk transform consists exactly of odd functions on $ S^{n-1} $, which is essential for characterizing the null space.
• A function $ f \in L_{\text{even}}^1(S^{n-1}) $ is constant if and only if its Minkowski-Funk transform is constant, as shown via spherical harmonic decomposition.
• The generalized transform $ M_\theta f $ provides a continuous family of transforms interpolating between the standard Minkowski-Funk transform and the spherical average $ M^t f $.
• The result $ U(\omega) = \frac{1}{2} (M B)(\omega) $, where $ U $ is the surface area measure and $ B $ is the support function, leads directly to Minkowski’s existence theorem for convex bodies.