Super-recursive algorithm

The Minkowski content of a set (named after Hermann Minkowski), or the boundary measure, is a basic concept in geometry and measure theory which generalizes to arbitrary measurable sets the notions of length of a smooth curve in the plane and area of a smooth surface in the space. It is typically applied to fractal boundaries of domains in the Euclidean space, but makes sense in the context of general metric measure spaces. It is related to, although different from, the Hausdorff measure.

Definition

Let $\scriptstyle (X,\,\mu ,\,d)$ be a metric measure space, where d is a metric on X and μ is a Borel measure. For a subset A of X and real ε > 0, let

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

be the ε-extension of A. The lower Minkowski content of A is given by

$M_{*}(A)=\liminf _{\varepsilon \to 0}{\frac {\mu (A_{\varepsilon })-\mu (A)}{2\varepsilon }},$

and the upper Minkowski content of A is

$M^{*}(A)=\limsup _{\varepsilon \to 0}{\frac {\mu (A_{\varepsilon })-\mu (A)}{2\varepsilon }}.$

If M_*(A) = M^*(A), then the common value is called the Minkowski content of A associated with the measure μ, and is denoted by M(A).

Minkowski content in Rⁿ

Let A be a subset of Rⁿ. Then the m-dimensional Minkowski content of A is defined as follows. The lower content is

$M_{*}^{m}(A)=\liminf _{\varepsilon \to 0}{\frac {\mu (A_{\varepsilon })-\mu (A)}{\alpha _{n-m}\varepsilon ^{n-m}}}$

where α_n−m is the volume of the unit (n−m)-ball and $\mu$ is $n$-dimensional Lebesgue measure. The upper content is

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

As before, if the upper and lower m-dimensional Minkowski content of A agree, then the Minkowski content of A, M^m(A), is defined to be this common value.

Properties

• The Minkowski content is (generally) not a measure. In particular, the m-dimensional Minkowski content in Rⁿ is not a measure unless m = 0, in which case it is the counting measure. Indeed, clearly the Minkowski content assigns the same value to the set A as well as its closure.
• If A is a closed m-rectifiable set in Rⁿ, given as the image of a bounded set from R^m under a Lipschitz function, then the m-dimensional Minkowski content of A exists, and is equal to the m-dimensional Hausdorff measure of A, apart from a constant normalization depending on the dimension.

See also

• Gaussian isoperimetric inequality
• Geometric measure theory
• Isoperimetric problem
• Minkowski–Bouligand dimension

References

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• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
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