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 n ≥ 2 {\displaystyle n\geq 2} , and let A ⊊ R n {\displaystyle A\subsetneq \mathbb {R} ^{n}} be a compact set. Let μ ( A ) {\displaystyle \mu (A)} denote the Lebesgue measure (volume) of A {\displaystyle A} . Define the quantity λ ( ∂ A ) {\displaystyle \lambda (\partial A)} by the Minkowski–Steiner formula
λ ( ∂ A ) := lim inf δ → 0 μ ( A + B δ ¯ ) − μ ( A ) δ , {\displaystyle \lambda (\partial A):=\liminf _{\delta \to 0}{\frac {\mu \left(A+{\overline {B_{\delta }}}\right)-\mu (A)}{\delta }},}
where
B δ ¯ := { x = ( x 1 , … , x n ) ∈ R n | | x | := x 1 2 + ⋯ + x n 2 ≤ δ } {\displaystyle {\overline {B_{\delta }}}:=\left\{x=(x_{1},\dots ,x_{n})\in \mathbb {R} ^{n}\left||x|:={\sqrt {x_{1}^{2}+\dots +x_{n}^{2}}}\leq \delta \right.\right\}}
denotes the closed ball of radius δ > 0 {\displaystyle \delta >0} , and
A + B δ ¯ := { a + b ∈ R n | a ∈ A , b ∈ B δ ¯ } {\displaystyle A+{\overline {B_{\delta }}}:=\left\{a+b\in \mathbb {R} ^{n}\left|a\in A,b\in {\overline {B_{\delta }}}\right.\right\}}
is the Minkowski sum of A {\displaystyle A} and B δ ¯ {\displaystyle {\overline {B_{\delta }}}} , so that
A + B δ ¯ = { x ∈ R n | | x − a | ≤ δ for some a ∈ A } . {\displaystyle A+{\overline {B_{\delta }}}=\left\{x\in \mathbb {R} ^{n}{\mathrel {|}}\ {\mathopen {|}}x-a{\mathclose {|}}\leq \delta {\mbox{ for some }}a\in A\right\}.}
Remarks
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