Generalized averages of section and projection functions

In this paper, we discuss the determination of a convex or star-shaped body K in Rd by information about the sizes of sections or projections. Following the work of Groemer [H. Groemer, On a spherical integral transform and sections of star bodies, Monatsh. Math. 126 (1998) 117–124], we considered in [P. Goodey, W. Weil, Average section functions for star-shaped sets, Adv. in Appl. Math. 36 (2006) 70–84] directed section functions sk(K;⋅), describing the content of the intersection K∩H of K with k-dimensional half-spaces H, 1⩽k⩽d−1. We showed that sk(K;⋅) determines the body K uniquely, whereas, for the integrals of sk(K;⋅) over all half-spaces H containing a given (normal) direction, uniqueness only holds for certain pairs (k,d).Here, we study a more general situation and consider, for 2⩽j⩽k⩽d−1, the averages of sk(K;⋅) over all half-spaces H containing a fixed j-dimensional half-space G with inner normal u. We show that the resulting function , of the variables G and u, determines K uniquely. This is in contrast to our earlier result which concerned . We also extend this uniqueness to the case k