Talk:Minkowski–Bouligand dimension

Initial one
Hmmm ... there should be a page on this, but the initial one consisted of nothing but (interesting-looking) references. Charles Matthews 18:10, 19 May 2004 (UTC)
 * I've added in a definition, but am not so familiar with the fractals side of things and so it still needs more work. Also one may need to set up a whole bunch of redirects... Terry 17:59, 14 Oct 2004 (UTC)
 * Ack. I've discovered that there is a lot of overlap between what I just wrote here and the article on box-counting dimension.  Now we need to merge... sigh Terry 18:05, 14 Oct 2004 (UTC)

merged from Talk:Box-counting dimension:

Box-counting dimension is not identical with Hausdorff dimension; this redirect is inappropriate, therefore.Charles Matthews 18:01, 27 Feb 2004 (UTC)
 * It is, however, identical with Minkowski-Bouligand dimension, which also goes under a variety of other names... we should probably merge these two articles now. Terry 18:09, 14 Oct 2004 (UTC)
 * I'll do it, I hope I might have time the weekend. Gadykozma 18:53, 14 Oct 2004 (UTC)
 * I had a crack at it already... but it might still be worth someone else looking over it and making sure it is OK. Terry 03:12, 15 Oct 2004 (UTC)
 * Very nice job. Thanks. Gadykozma 00:55, 16 Oct 2004 (UTC)

I would like to add that the packing dimension is different from the Minkowski dimension, I refer you to chapter 3.4 of Falconers "Fractal Geometry; Mathematical Foundations and Applications". I will edit this tommorow if I have the time. Andy

I've redirected "box dimension" here. --Druiffic (talk) 17:13, 19 December 2008 (UTC)Druiffic

Non-bounded sets
Can this dimension be defined for unbounded sets? —Preceding unsigned comment added by 90.229.231.115 (talk) 18:56, 30 January 2009 (UTC)

Category: Entropy?
Removing from category:Entropy. Unlike fractal dimension, this article doesn't even mention Renyi entropy. -- Jheald 15:28, 6 March 2007 (UTC)

Packing dimension
Packing dimension is not the same as Minkowski dimension. Packing dimension of a set A is the infimum of all numbers s for which P(A,ε)εs converges to zero where P(A,ε) is the packing number. For general sets, the Hausdorff dimension is at most the Minkowski dimension which is at most the packing dimension, but no two have to be equal for any set. For self-similar sets, they are identical, I believe, but I'll need to check a reference. Jazzam 16:50, 26 June 2007 (UTC)

Actually, my bad. The packing number can be used to compute the Minkowski dimension. I meant to say that there is an actual dimension called packing dimension distinct from Minkowski and Hausdorff dimension, where the inequalities mentioned above are still true, but it is constructed in a different manner. The way you define it is as follows: the upper packing dimensions are, respectively,
 * $$ \overline{\mbox{dim}}_{P}(A)=\inf\{\sup_{i}\overline{\mbox{dim}}_{M}A_{i}:A=\bigcap A_{i}\}$$

and the $$A_{i}$$ are bounded. Lower packing dimension is defined similarly, only using lower Minkowski dimension instead of upper. Sorry for the confusion about how to construct it. My reference is Mattila's "Geometry of Sets and Measures." Also, I said before that the dimensions may all be equal for self-similar sets. This is true as long as the self-similar set satisfies the open set condition. Jazzam 18:48, 26 June 2007 (UTC)

error
The lower box dimension is not finitely additive. A counterexample can be constructed from the set that is already used to show that there are sets for which upper and lower box dimension do not yield the same value. —Preceding unsigned comment added by 78.54.85.157 (talk) 17:15, 11 November 2009 (UTC)

Relations to the Hausdorff dimension: LaTeX needs fixing
The formulas in the "Relations to the Hausdorff dimension" section are showing an error message " ," at least when viewed using Nageh's MathJax. I had a quick glance at the code but didn't know enough about LaTeX to see what was wrong. SoledadKabocha (talk) 00:53, 23 August 2012 (UTC)

Alternative definitions: unclear
"covering number is the minimal number of open balls" – OK;

"packing number is the maximal number of disjoint balls" – open balls or closed balls? Boris Tsirelson (talk) 17:52, 26 November 2013 (UTC)

In fact, in order to have
 * $$ N'_\text{covering}(2\varepsilon) \leq N_\text{covering}(\varepsilon), \quad N_\text{packing}(\varepsilon) \leq N'_\text{covering}(\varepsilon) \, $$

one needs open balls in one definition and closed balls in the other. There are two possibilities, and both are used by some authors. Given that in this article open balls are used for covering, closed balls must be used for packing. I insert it into the article. Boris Tsirelson (talk) 19:58, 26 November 2013 (UTC)

Oops! No, sorry, I was not careful enough. One may use a strict inequality when defining the covering number and a non-strict inequality when defining the packing number. Or the other way round. But what about the balls?

When defining the covering number, strict inequality means open balls (while non-strict inequality means closed balls).

But when defining the packing number, non-strict inequality (distance more or equal 2r between two centers) means two disjoint open balls. And strict inequality (distance more than 2r between two centers) means two disjoint closed balls.

Thus: either only open balls, or only closed balls.

This is the situation in the Euclidean space. In a general metric space it may happen that two r-balls are disjoint even though they centers are closer than 2r. Boris Tsirelson (talk) 11:20, 27 November 2013 (UTC)