  We study the structures of two types of generalizations of
intersection-bodies and the problem of whether they are in fact equivalent.
Intersection-bodies were introduced by Lutwak and played a key role in the
solution of the Busemann-Petty problem. A natural geometric generalization of
this problem considered by Zhang, led him to introduce one type of generalized
intersection-bodies. A second type was introduced by Koldobsky, who studied a
different analytic generalization of this problem. Koldobsky also studied the
connection between these two types of bodies, and noted that an equivalence
between these two notions would completely settle the unresolved cases in the
generalized Busemann-Petty problem. We show that these classes share many
identical structure properties, proving the same results using Integral
Geometry techniques for Zhang's class and Fourier transform techniques for
Koldobsky's class. Using a Functional Analytic approach, we give several
surprising equivalent formulations for the equivalence problem, which reveal a
deep connection to several fundamental problems in the Integral Geometry of the
Grassmann Manifold.
