Cauchy's surface area formula states that the surface area 
of a compact convex set 
with nonempty interior is a constant multiple of the
average 
-dimensional volume
of its orthogonal projections. More precisely,
 |
where 
ranges over the unit sphere, 
is the orthogonal
projection onto the hyperplane perpendicular
to 
,
is 
-dimensional Lebesgue measure,
and 
is the volume of the 
-dimensional unit
ball.
For 
,
this becomes Cauchy's perimeter formula.
For 
,
it says that the surface area is four times the average area of the orthogonal
projections.
