Minkowski capacity

Minkowski capacity is a basic concept in geometric measure theory , generalizing to arbitrary measurable sets the concepts of the length of a curve on a plane and the surface area in space .

Capacity is usually used for fractal boundaries of regions in Euclidean space , but it makes sense in the context of general metric spaces with measure.

Named after German Minkowski .

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Definition

Let be $(X,\mu ,d)$ ${\ displaystyle (X, \ mu, d)}$ metric space with measure where $d$ ${\ displaystyle d}$ is a metric on $X$ ${\ displaystyle X}$ , but $\mu$ ${\ displaystyle \ mu}$ Is a Borel measure . For a subset $A$ ${\ displaystyle A}$ at $X$ ${\ displaystyle X}$ and real ε> 0, we denote

A_{\varepsilon }=\{x\in X\,|\,d(x,A)<\varepsilon \}

{\ displaystyle A _ {\ varepsilon} = \ {x \ in X \, | \, d (x, A) <\ varepsilon \}}

its closed $\varepsilon$ ${\ displaystyle \ varepsilon}$ neighborhood. Lower capacity of Minkowski codimension $k$ ${\ displaystyle k}$ defined as lower limit

M_{*}(A)=\liminf _{\varepsilon \to 0}{\frac {\mu (A_{\varepsilon })-\mu (A)}{\varepsilon ^{k}}},

{\ displaystyle M _ {*} (A) = \ liminf _ {\ varepsilon \ to 0} {\ frac {\ mu (A _ {\ varepsilon}) - \ mu (A)} {\ varepsilon ^ {k}}} ,}

and upper capacity of Minkowski codimension $k$ ${\ displaystyle k}$ as the upper limit

M^{*}(A)=\limsup _{\varepsilon \to 0}{\frac {\mu (A_{\varepsilon })-\mu (A)}{\varepsilon ^{k}}}.

{\ displaystyle M ^ {*} (A) = \ limsup _ {\ varepsilon \ to 0} {\ frac {\ mu (A _ {\ varepsilon}) - \ mu (A)} {\ varepsilon ^ {k}} }.}

If a $M^{*}(A)=M_{*}(A)$ ${\ displaystyle M ^ {*} (A) = M _ {*} (A)}$ , then their general value is called the Minkowski capacity of codimension $k$ ${\ displaystyle k}$ A in measure μ, and is denoted by $M(A)$ ${\ displaystyle M (A)}$ .

Properties

If a $A$ ${\ displaystyle A}$ there is a closed $n$ ${\ displaystyle n}$ - rectifiable set in $\mathbb {R} ^{n+k}$ ${\ displaystyle \ mathbb {R} ^ {n + k}}$ , then the Minkowski capacity $A$ ${\ displaystyle A}$ in relation to volume on $\mathbb {R} ^{n+k}$ ${\ displaystyle \ mathbb {R} ^ {n + k}}$ exists and coincides with his $n$ ${\ displaystyle n}$ -Hausdorff measure up to normalization.

Links

Federer, Herbert , Geometric measure theory .

Minkowski capacity

Content

Definition

Properties

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