The Mandelbrot set has a well defined area. Its boundary has Hausdorff dimension 2, so what is its perimeter?

Question

More specifically, is it finite or infinite? My understanding is that Hausdorff dimension is a way to make bounded sets have a finite "volume", but I'm not sure about the Mandelbrot Set.

Answer 1

"Perimeter" means "length of the boundary". "Length" refers to one-dimensional measure - since the boundary has dimension greater than $1$, the boundary's one-dimensional measure is necessarily infinite (by the definition of Hausdorff dimension, the $s$-dimensional measure is infinite for $s$ less than the Hausdorff dimension).

Now, the question you might be asking is what the area of the boundary is ("area" refers to two-dimensional measure, which is the only Hausdorff measure that could in principle be nonzero and noninfinite in this case). That actually has been discussed on MathOverflow (https://mathoverflow.net/questions/37229/area-of-the-boundary-of-the-mandelbrot-set) with the conclusion that it's still an open question whether that area is positive, let alone what its actual value is. That was back in 2010, so the field has had nearly eight years to develop from there - but a quick literature search turns up nothing, so this is probably still an open question.