Definition:Inner Jordan Content

From ProofWiki
Jump to navigation Jump to search

Definition

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


Let:

$\ds R = \prod_{i \mathop = 1}^n \closedint {a_i} {b_i}$

be a closed $n$-rectangle that contains $M$.

Let $V_R \in \R_{\ge 0}$ be defined as:

$\ds V_R = \prod_{i \mathop = 1}^n \paren {b_i - a_i}$


The inner Jordan content of $M$ is defined and denoted as:

$\map {m_*} M = V_R - \map {m^*} {R \setminus M}$

where:

$\map {m^*} {R \setminus M}$ denotes the outer Jordan content of $R \setminus M$


That is, the inner Jordan content of $M$ is defined to be the outer Jordan content of the complement of $M$, relative to some fixed $R \supseteq M$.


This article, or a section of it, needs explaining.
In particular: It appears that we have to pick some arbitrary $R$ before we can define the inner Jordan content. Hence it needs to be clarified that it doesn't matter what the exact details of $R$ are, as long as it contains $M$, the inner Jordan content is the same whatever $R$ is. Is this correct? If so, we need to add a section explaining this, because it is not obvious. Presumably Inner Jordan Content is Well-Defined is going to take on the job of proving it, but it's still worth adding the note of explanation.
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.



Also see

  • Results about inner Jordan content can be found here.


Source of Name

This entry was named for Marie Ennemond Camille Jordan.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Inner_Jordan_Content&oldid=692676"