Generalized averages of section and projection functions

Abstract

In this paper, we discuss the determination of a convex or star-shaped body K in R d 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 s k ( K ; ⋅ ) , describing the content of the intersection K ∩ H of K with k-dimensional half-spaces H, 1 ⩽ k ⩽ d − 1 . We showed that s k ( K ; ⋅ ) determines the body K uniquely, whereas, for the integrals of s k ( 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 s k ( K ; ⋅ ) over all half-spaces H containing a fixed j-dimensional half-space G with inner normal u. We show that the resulting function s ¯ j k ( K ; ⋅ ) , of the variables G and u, determines K uniquely. This is in contrast to our earlier result which concerned s ¯ 1 k ( K ; ⋅ ) . We also extend this uniqueness to the case k < j . Similar results are obtained for projections of convex bodies.