Average section functions for star-shaped sets

In [H. Groemer, On a spherical integral transform and sections of star bodies, Monatsh. Math. 126 (1998) 117–124], Groemer considered intersections of star bodies K⊂Rd, d⩾3, (with respect to the origin 0) with (d−1)-dimensional half-spaces through 0 and showed that the ((d−1)-dimensional) volumes of these intersections determine the body K uniquely. A direct consequence is that a corresponding uniqueness result also holds for intersections with j-dimensional half-spaces, for any fixed j∈{2,…,d−1}. The result in [H. Groemer, On a spherical integral transform and sections of star bodies, Monatsh. Math. 126 (1998) 117–124] is the outcome of combining injectivity results for two spherical transforms, the hemispherical transform and the (spherical) Radon transform. Here, we introduce the average sj(K,u) of the intersection volume over all j-dimensional half-spaces H containing a given direction u (orthogonal to the boundary of H). It is a more general question whether the resulting function sj(K,⋅) on the unit sphere Sd−1 determines K uniquely. In fact, we shall show uniqueness in the cases j=2 and d=3 or 4, as well as j⩽(d+2)/2 and j>(2d+1)/3, for d⩾5, and we give corresponding stability results. On the other hand, we shall see that there are infinitely many pairs (j,d) for which uniqueness fails.