The topological type of the α-sections of convex sets

An α=(α1,…,αk)(0⩽αi⩽1) section of a family {K1,…,Kk} of convex bodies in Rd is a transversal halfspace H+ for which Vold(Ki∩H+)=αi⋅Vold(Ki) for every 1⩽i⩽k. Our main result is that for any well-separated family of strictly convex sets, the space of α-sections is diffeomorphic to Sd−k.