A Fourier transform approach to Christoffel’s problem

Abstract

Abstract. We use Fourier transform techniques to provide a new approach to Berg's solution of the Christoffel problem. This leads to an explicit description of Berg's spherical kernel and to new regularity properties of the associated integral transform.

Even Minkowski valuations

“…Goodey and Weil [23] also determined the family of generating functions for the mean section operators. In order to explain their result, recall that in Berg's solution of the Christoffel-Minkowski problem (see, e.g., [25,57]) he proved the following: For every n ≥ 2 there exists a uniquely determined C ∞ function ζ n on (−1, 1) such that the associated zonal functionζ n ∈ L 1 (S n−1 ) is orthogonal to the restriction of all linear functions to S n−1 and satisfies, for every K ∈ K n , h(JK, ·) = S 1 (K, ·) * ζ n . (5.11)…”

Projection functions, area measures and the Alesker–Fourier transform

“…Since p > −n, f p is locally integrable and determines an even homogeneous distribution of degree p acting on test functions by integration. Thus, by Lemma 2.4, Ff p is an even homogeneous distribution of degree −n − p. It was first noted in [33] that, for −n < p < 0, Ff p is, in fact, an infinitely differentiable function on R n \{0} (which is even and homogeneous of degree −n − p). This gives rise to an operator F p on C ∞ e (S n−1 ), called the spherical Fourier transform of degree p ∈ (−n, 0), defined by…”

“…From the computation of the multipliers of the Alesker-Fourier transform of spherical valuations in [14] and the spherical Fourier transform in [33], it follows that Corollary 3.9 also holds without the assumption on the parity.…”

“…Clearly, F p is a linear and SO(n) equivariant map. Hence, by Schur's lemma, F p acts as a multiplier transformation on the spaces H n 2k , k ∈ N. Its multipliers a n 2k [F p ] were determined in [33] and are given by…”

Existence of solutions to the even dual Minkowski problem

“…A short approach to their result can be found in Grinberg & Zhang [22]. A Fourier transform approach can be found in Goodey, Yaskin & Yaskina [21]. For polytopes, a direct treatment was given by Schneider [54].…”