Covering shadows with a smaller volume

Abstract: For a construction is given for convex bodies K and L in such that the orthogonal projection onto the subspace contains a translate of for every direction u, while the volumes of K and L satisfy . A more general construction is then given for n-dimensional convex bodies K and L such that each orthogonal projection onto a k-dimensional subspace ξ contains a translate of , while the mth intrinsic volumes of K and L satisfy for all . For each , we then define the collection to be the closure (under the Hausdorff topology) of all Blaschke combinations of suitably defined cylinder sets (prisms). It is subsequently shown that, if , and if the orthogonal projection contains a translate of for every k-dimensional subspace ξ of , then . The families , called k-cylinder bodies of , form a strictly increasing chain where is precisely the collection of centrally symmetric compact convex sets in , while is the collection of all compact convex sets in . Members of each family are seen to play a fundamental role in relating covering conditions for projections to the theory of mixed volumes, and members of are shown to satisfy certain geometric inequalities. Related open questions are also posed. [Copyright &y& Elsevier]