Cauchy's perimeter formula states that the perimeter 
of the boundary of a compact convex set 
with nonempty interior in the
plane is the integral of its
orthogonal projection lengths
over all unoriented directions. If 
is the length of the orthogonal projection of 
onto a line whose direction
makes angle 
with a fixed axis, then
 |
Equivalently, 
is the distance between the two supporting lines perpendicular to that direction.
In particular, if this distance has the constant value
,
the formula gives 
, which is Barbier's theorem.
Cauchy's perimeter formula is the two-dimensional case of Cauchy's surface area formula.
