Minkowski–Steiner formula

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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.

The Minkowski–Steiner formula is used, together with the Brunn–Minkowski theorem, to prove the isoperimetric inequality. It is named after Hermann Minkowski and Jakob Steiner.

Statement of the Minkowski-Steiner formula[edit]

Let , and let be a compact set. Let denote the Lebesgue measure (volume) of . Define the quantity by the Minkowski–Steiner formula

where

denotes the closed ball of radius , and

is the Minkowski sum of and , so that

Remarks[edit]

Surface measure[edit]

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.

Convex sets[edit]

When the set is a convex set, the lim-inf above is a true limit, and one can show that

where the are some continuous functions of (see quermassintegrals) and denotes the measure (volume) of the unit ball in :

where denotes the Gamma function.

Example: volume and surface area of a ball[edit]

Taking gives the following well-known formula for the surface area of the sphere of radius , :

where is as above.

References[edit]

  • Dacorogna, Bernard (2004). Introduction to the Calculus of Variations. London: Imperial College Press. ISBN 1-86094-508-2. 
  • Federer, Herbert (1969). Geometric Measure Theory. New-York: Springer-Verlag.