A cylinder covering of a set 
is a finite family of
generalized cylinders 
whose union contains 
. Each 
is a unit vector, and its
base 
is a measurable set of finite area. All norms and areas
use the Euclidean convention. The bases need not be disks.
The cost is 
,
counting overlaps with multiplicity (Bezdek and Litvak 2009). For a convex
body 
,
write 
for orthogonal projection and define
 |
The half-area conjecture proposes that every such covering has cost at least 
. The stronger directionwise conjecture
proposes
 |
The directionwise ratios are invariant under invertible affine transformations. They are different from a ratio using one minimum projection area for all directions.
