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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.

Contents

Statement of the Minkowski-Steiner formulaEdit

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

 

RemarksEdit

Surface measureEdit

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 setsEdit

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 ballEdit

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

 
 
 

where   is as above.

ReferencesEdit

  • 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.