Spherical projections and liftings in geometric tomography

Abstract

We consider a variety of integral transforms arising in Geometric Tomography. It will be shown that these can be put into a common framework using spherical projection and lifting operators. These operators will be applied to support functions and surface area measures of convex bodies and to radial functions of star bodies. We then investigate averages of lifted projections and show that they correspond to self-adjoint intertwining operators. We obtain formulas for the eigenvalues of these operators and use them to ascertain circumstances under which tomographic measurements determine the original bodies. This approach via mean lifted projections leads us to some unexpected relationships between seemingly disparate geometric constructions.

Kinematic formulas for area measures

“…Remark 4.3. Although we mentioned in the introduction that a Crofton formula with integrand S ′ j (K ∩E, •), the area measure of K ∩E calculated in the subspace L(E), is not possible in a direct manner, we can obtain a version of (3.6) with S j (K ∩E, •) replaced by S ′ j (K ∩E, •) if we use an appropriate lifting for measures from the unit sphere in L(E) to S d−1 (see [5]). Namely, it was shown in [5, Theorem 6.2] that…”

Large faces in Poisson hyperplane mosaics

“…for Borel sets A ⊂ S k−1 L and for µ ∈ M(S d−1 ). (More general spherical projections and their applications are treated in [3]. )…”