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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

Let n2, and let An be a compact set. Let μ(A) denote the Lebesgue measure (volume) of A. Define the quantity λ(A) by the Minkowski–Steiner formula

λ(A):=lim infδ0μ(A+Bδ)μ(A)δ,

where

Bδ:={x=(x1,,xn)n||x|:=x12++xn2δ}

denotes the closed ball of radius δ>0, and

A+Bδ:={a+bn|aA,bBδ}

is the Minkowski sum of A and Bδ, so that

A+Bδ={xn||xa|δ for some aA}.

Remarks

Surface measure

For "sufficiently regular" sets A, the quantity λ(A) does indeed correspond with the (n1)-dimensional measure of the boundary A of A. See Federer (1969) for a full treatment of this problem.

Convex sets

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

μ(A+Bδ)=μ(A)+λ(A)δ+i=2n1λi(A)δi+ωnδn,

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

ωn=2πn/2nΓ(n/2),

where Γ denotes the Gamma function.

Example: volume and surface area of a ball

Taking A=BR gives the following well-known formula for the surface area of the sphere of radius R, SR:=BR:

λ(SR)=limδ0μ(BR+Bδ)μ(BR)δ
=limδ0[(R+δ)nRn]ωnδ
=nRn1ωn,

where ωn is as above.

References

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  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534