How to calculate Minkowski dimension of the set bounded with $x=0$, $y=0$; $x+y=1$?
Which metric should I use?
I tried with Chebyshev metric but I have problem on the edges of the triangle.
Calculate the Minkowski dimension of a triangle
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fractals
dimension-theory
1 Answers
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A metric should be chosen before the question "what is the Minkowski dimension" is raised; otherwise the question is under-specified and has no answer. Abstract sets do not have the concept of Minkowski dimension.
When no metric on a subset of $\mathbb{R}^n$ is indicated, one is to assume that the Euclidean metric is used. So, use that. The dimension is expected to be $2$, since the Minkowski dimension agrees with the intuitive notion of dimension for objects that look like a curve, or a plane, etc. To confirm this expectation:
- Cover the triangle by $N^2$ small squares of sidelength $1/N$, obtained by uniformly slicing the square $[0,1]\times [0,1]$. This gives the upper bound $\overline{\dim_M} \le 2$.
- Observe that the triangle contains the following points separated by distance $\ge \delta$ from one another: $\{(i\delta, j\delta): i,j=0,1,\dots, N\}$, where $N = \lfloor 1/(2\delta)\rfloor $. So, any cover of the triangle by sets of diameter less than $\delta$ will consist of at least $N^2$ sets. This leads to lower bound $\underline{\dim_M} \ge 2$.