Category:Definitions/Outer Jordan Content

This category contains definitions related to Outer Jordan Content.
Related results can be found in Category:Outer Jordan Content.

Let $M \subseteq \R^n$ be a bounded subspace of Euclidean $n$-space.

The outer Jordan content of $M$ is defined as denoted as:

$\ds \map {m^*} M = \inf_C \sum_{R \mathop \in C} \map V R$

where:

the infimum $\ds \inf_C$ is taken over all finite coverings $C$ of $M$ by closed $n$-rectangles

This article, or a section of it, needs explaining. In particular: The ordered set and ordering relation need to be clarified. Presumably this is the "set of all finite coverings of $M$" and the subset relation. Does it need to be proved that the "set of all finite coverings of $M$" actually admits an infimum? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

$R = \closedrect {\mathbf a} {\mathbf b}$ denotes a closed $n$-rectangle of $C$ for $\mathbf a, \mathbf b \in \R^n$

$\map V R$ is defined as being:

$\map V R := \ds \prod_{i \mathop = 1}^n \paren {b_i - a_i}$

where $a_i$, $b_i$ are the coordinates of $\mathbf a = \tuple {a_1, a_2, \dotsc, a_n}$ and $\mathbf b = \tuple {b_1, b_2, \dotsc, b_n}$.