In mathematics, the Minkowski–Steiner formula is a formula relating the surface area and volume of compact subsets of Euclidean space. More precisely, it defines the surface area as the "derivative" of enclosed volume in an appropriate sense.
Statement of the Minkowski-Steiner formula
Let , and let be a compact set. Let denote the Lebesgue measure (volume) of . Define the quantity by the Minkowski–Steiner formula
is the Minkowski sum of and , so that
For "sufficiently regular" sets , the quantity does indeed correspond with the -dimensional measure of the boundary of . See Federer (1969) for a full treatment of this problem.
where denotes the Gamma function.
Example: volume and surface area of a ball
Taking gives the following well-known formula for the surface area of the sphere of radius , :
where is as above.