Minkowski–Steiner formula

Let $n\; \backslash geq\; 2$, and let $A\; \backslash subsetneq\; \backslash mathbb\{R\}^\{n\}$ be a compact set. Let $\backslash mu\; (A)$ denote the Lebesgue measure (volume) of $A$. Define the quantity $\backslash lambda\; (\backslash partial\; A)$ by the Minkowski–Steiner formula

:$\backslash lambda\; (\backslash partial\; A)\; :=\; \backslash liminf\_\{\backslash delta\; \backslash to\; 0\}\; \backslash frac\{\backslash mu\; \backslash left(\; A\; +\; \backslash overline\{B\_\{\backslash delta\}\}\; \backslash right)\; -\; \backslash mu\; (A)\}\{\backslash delta\},$

where

:$\backslash overline\{B\_\{\backslash delta\}\}\; :=\; \backslash left\backslash \{\; x\; =\; (x\_\{1\},\; \backslash dots,\; x\_\{n\})\; \backslash in\; \backslash mathbb\{R\}^\{n\}\; \backslash left|\; |\; x\; |\; :=\; \backslash sqrt\{x\_\{1\}^\{2\}\; +\; \backslash dots\; +\; x\_\{n\}^\{2\}\}\; \backslash leq\; \backslash delta\; \backslash right.\; \backslash right\backslash \}$

denotes the closed ball of radius $\backslash delta\; >\; 0$, and

:$A\; +\; \backslash overline\{B\_\{\backslash delta\}\}\; :=\; \backslash left\backslash \{\; a\; +\; b\; \backslash in\; \backslash mathbb\{R\}^\{n\}\; \backslash left|\; a\; \backslash in\; A,\; b\; \backslash in\; \backslash overline\{B\_\{\backslash delta\}\}\; \backslash right.\; \backslash right\backslash \}$

is the Minkowski sum of $A$ and $\backslash overline\{B\_\{\backslash delta\}\}$, so that

:$A\; +\; \backslash overline\{B\_\{\backslash delta\}\}\; =\; \backslash left\backslash \{\; x\; \backslash in\; \backslash mathbb\{R\}^\{n\}\; \backslash mathrel|\backslash \; \backslash mathopen|\; x\; -\; a\; \backslash mathclose|\; \backslash leq\; \backslash delta\; \backslash mbox\{\; for\; some\; \}\; a\; \backslash in\; A\; \backslash right\backslash \}.$

For "sufficiently regular" sets $A$, the quantity $\backslash lambda\; (\backslash partial\; A)$ does indeed correspond with the $(n\; -\; 1)$-dimensional measure of the boundary $\backslash partial\; A$ of $A$. See Federer (1969) for a full treatment of this problem.

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

:$\backslash mu\; \backslash left(\; A\; +\; \backslash overline\{B\_\{\backslash delta\}\}\; \backslash right)\; =\; \backslash mu\; (A)\; +\; \backslash lambda\; (\backslash partial\; A)\; \backslash delta\; +\; \backslash sum\_\{i\; =\; 2\}^\{n\; -\; 1\}\; \backslash lambda\_\{i\}\; (A)\; \backslash delta^\{i\}\; +\; \backslash omega\_\{n\}\; \backslash delta^\{n\},$

where the $\backslash lambda\_\{i\}$ are some continuous functions of $A$ (see quermassintegrals) and $\backslash omega\_\{n\}$ denotes the measure (volume) of the unit ball in $\backslash mathbb\{R\}^\{n\}$:

:$\backslash omega\_\{n\}\; =\; \backslash frac\{2\; \backslash pi^\{n\; /\; 2\}\}\{n\; \backslash Gamma\; (n\; /\; 2)\},$

where $\backslash Gamma$ denotes the Gamma function.

Taking $A\; =\; \backslash overline\{B\_\{R\}\}$ gives the following well-known formula for the surface area of the sphere of radius $R$, $S\_\{R\}\; :=\; \backslash partial\; B\_\{R\}$:

:$\backslash lambda\; (S\_\{R\})\; =\; \backslash lim\_\{\backslash delta\; \backslash to\; 0\}\; \backslash frac\{\backslash mu\; \backslash left(\; \backslash overline\{B\_\{R\}\}\; +\; \backslash overline\{B\_\{\backslash delta\}\}\; \backslash right)\; -\; \backslash mu\; \backslash left(\; \backslash overline\{B\_\{R\}\}\; \backslash right)\}\{\backslash delta\}$

::$=\; \backslash lim\_\{\backslash delta\; \backslash to\; 0\}\; \backslash frac\{[\; (R\; +\; \backslash delta)^\{n\}\; -\; R^\{n\}\; ]\; \backslash omega\_\{n\}\}\{\backslash delta\}$

::$=\; n\; R^\{n\; -\; 1\}\; \backslash omega\_\{n\},$

where $\backslash omega\_\{n\}$ is as above.

• {{cite book | author=Dacorogna, Bernard | title=Introduction to the Calculus of Variations | publisher=Imperial College Press | location=London | year=2004 | isbn=1-86094-508-2 }}
• {{cite book | author=Federer, Herbert | title=Geometric Measure Theory | publisher=Springer-Verlag | location=New-York | year=1969 }}

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