Covering shadows with a smaller volume

Abstract

For n ⩾ 2 a construction is given for convex bodies K and L in R n such that the orthogonal projection L u onto the subspace u ⊥ contains a translate of K u for every direction u, while the volumes of K and L satisfy V n ( K ) > V n ( L ) . A more general construction is then given for n-dimensional convex bodies K and L such that each orthogonal projection L ξ onto a k-dimensional subspace ξ contains a translate of K ξ , while the mth intrinsic volumes of K and L satisfy V m ( K ) > V m ( L ) for all m > k . For each k = 1 , … , n , we then define the collection C n , k to be the closure (under the Hausdorff topology) of all Blaschke combinations of suitably defined cylinder sets (prisms). It is subsequently shown that, if L ∈ C n , k , and if the orthogonal projection L ξ contains a translate of K ξ for every k-dimensional subspace ξ of R n , then V n ( K ) ⩽ V n ( L ) . The families C n , k , called k-cylinder bodies of R n , form a strictly increasing chain C n , 1 ⊂ C n , 2 ⊂ ⋯ ⊂ C n , n − 1 ⊂ C n , n , where C n , 1 is precisely the collection of centrally symmetric compact convex sets in R n , while C n , n is the collection of all compact convex sets in R n . Members of each family C n , k are seen to play a fundamental role in relating covering conditions for projections to the theory of mixed volumes, and members of C n , k are shown to satisfy certain geometric inequalities. Related open questions are also posed.