Fourier transforms and the Funk-Hecke theorem in convex geometry

Abstract

We apply Fourier transforms to homogeneous extensions of functions on S n−1 . This results in complex integral operators. The real and imaginary parts of these operators provide a pairing of stereological data that leads to new results concerning the determination of convex bodies as well as new settings for known results. Applying the Funk-Hecke theorem to these operators yields stability versions of the results.

A Fourier transform approach to Christoffel’s problem

“…for ξ ∈ S n−1 ; see [15], for example. For non-integer values of p ∈ C, the distributions |r| p−1 ∧ and |r| p−1 sgn r ∧ can be extended to an analytic family of distributions; see [11].…”

“…For non-integer values of p ∈ C, the distributions |r| p−1 ∧ and |r| p−1 sgn r ∧ can be extended to an analytic family of distributions; see [11]. In fact, regularization techniques can be used to extend them to many integer values of p. This was used in [15] to show, for example, that for ξ ∈ S n−1 ,…”

“…For these, we will make use of the equations (3.8) and (3.9) from [15]. If p is a non-positive even integer and f ∈ C ∞ S n−1 is even,…”

“…Thenf is also infinitely differentiable on R n \ o (see, for example, Corollary 3.3 of [15]) and it has homogeneity −n + 1. Thus |x| −2f e (x) is even and homogeneous of degree −n − 1, therefore, using the real part of (2), we obtain…”

“…For the vast majority of these applications, the transforms are restricted to even spherical functions. In [15], we extended these transform results to general spherical functions and this extension will play a key role in the current paper. These techniques will also allow us to establish new regularity results which, in turn, have implications for certain stability results.…”