Cauchy's projection formula gives the 
-dimensional volume
of the orthogonal projection of a compact convex set 
with nonempty interior
onto any hyperplane. Let 
be the measure on the unit
sphere induced by 
-dimensional boundary area under the Gauss map, which
sends almost every boundary point of 
to its outer unit normal vector.
For a unit vector 
, let 
be the hyperplane perpendicular
to 
.
Then
 |
where 
is the orthogonal projection and 
denotes 
-dimensional Lebesgue
measure.
Taking the average of this identity over 
gives Cauchy's surface area formula.
