The Minkowski

content Content or contents may refer to: Media * Content (media), information or experience provided to audience or end-users by publishers or media producers ** Content industry, an umbrella term that encompasses companies owning and providing mas ...

(named after

Hermann Minkowski Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician and professor at Königsberg, Zürich and Göttingen. He created and developed the geometry of numbers and used geometrical methods to solve problems in number t ...

), or the boundary measure, of a set is a basic concept that uses concepts from

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

and

measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...

to generalize the notions of

length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...

of a

smooth curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that a ...

in the plane, and

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...

of a smooth surface in

space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider ...

, to arbitrary

measurable set In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...

s. It is typically applied to

fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...

boundaries of domains in the

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

, but it can also be used in the context of general metric measure spaces. It is related to, although different from, the

Hausdorff measure In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that ass ...

Definition

For

A \subset \mathbb^

, and each integer ''m'' with

0 \leq m \leq n

, the ''m''-dimensional upper Minkowski content is :

M^(A) = \limsup_ \frac

and the ''m''-dimensional lower Minkowski content is defined as :

M_*^m(A) = \liminf_ \frac

where

\alpha(n-m)r^

is the volume of the (''n''−''m'')-ball of radius r and

\mu

is an

n

-dimensional

Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...

. If the upper and lower ''m''-dimensional Minkowski content of ''A'' are equal, then their common value is called the Minkowski content ''M''^''m''(''A'').

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 In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinity ...

. 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 mathematics, a rectifiable set is a set that is smooth in a certain measure-theoretic sense. It is an extension of the idea of a rectifiable curve to higher dimensions; loosely speaking, a rectifiable set is a rigorous formulation of a piece-wi ...

in R^''n'', given as the image of a bounded set from R^''m'' under a

Lipschitz function In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exis ...

, then the ''m''-dimensional Minkowski content of ''A'' exists, and is equal to the ''m''-dimensional

of ''A''.

Footnotes

References

* . * {{citation, first1=Steven G., last1=Krantz, first2=Harold R., last2=Parks, author2-link=Harold R. Parks, title=The geometry of domains in space, series=Birkhäuser Advanced Texts: Basler Lehrbücher, publisher=Birkhäuser Boston, Inc., publication-place=Boston, MA, year=1999, isbn=0-8176-4097-5, mr=1730695 . Measure theory Geometry Analytic geometry Dimension theory Dimension Measures (measure theory) Fractals Hermann Minkowski

Definition

Properties

See also

Footnotes

References