The projection body 
of a convex body 
, for 
, is the convex
body symmetric about the origin whose convex
body support function satisfies
 |
(1)
|
where 
is a unit vector and 
is orthogonal
projection onto the hyperplane perpendicular
to 
. Thus its defining function
records the volumes of the shadows of 
(Saroglou and Zvavitch 2017).
For the unit ball 
, the projection body is
 |
(2)
|
where 
is the volume of 
. The normalized projection
volume
 |
(3)
|
is invariant under invertible affine transformations. Petty's projection-volume conjecture proposes that ellipsoids minimize this quantity. Brannen's conjecture proposes that simplices maximize it. These statements concern the projection body's volume, rather than the volume of its polar body.
